Bernard Widrow Department of Electrical Engineering Stanford University
Eugene Walach IBM Israel Ltd.
Science & Technology
IEEE
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POWER ENGINEERING Mohamed E. El-Hawary, Series Editor
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WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication
Copyright 0 2008 by the Institute of Electrical and Electronics Engineers. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneouslyin Canada.
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We dedicate this work to our families over the generations. They helped us inspired us.
I would like to dedicate this book to my grandsons Jeffrey and Adam Skl to their parents Rick and Debbie, to my daughter Leslie, to my wife Ronna and to the memory of my parents Moe and Ida Widrow.
Bernard Wid
I would like to dedicate this book to my son Elad, to my daughter Algit my wife Rina, to my mother Sarah and to the memory of my father Benja Walach.
Eugene Wa
A Special Dedication to the Memory of Derrick Ngu
Derrick Nguyen completed the Ph.D. in Electrical Engineering at Stanford versity in June 1991. He was the first to develop neural controls for the “t backer-upper,” based on backpropagation through time. His work has wid plication in the field of nonlinear control. In his short life he accomplish great deal. He was a favorite of all who knew him.
Preface 1
2
The Adaptive Inverse Control Concept 1.O Introduction 1.1 Inverse Control 1.2 Sample Applications of Adaptive Inverse Control 1.3 An Outline or Road Map for This Book Bibliography Wiener Filters 2.0 Introduction 2.1 Digital Filters, Correlation Functions, z-Transforms 2.2 Two-sided (Unconstrained) Wiener Filters 2.3 Shannon-Bode Realization of Causal Wiener Filters 2.4 Summary Bibliography
3
Adaptive LMS Filters 3.0 Introduction 3.1 An Adaptive Filter 3.2 The Performance Surface 3.3 The Gradient and the Wiener Solution 3.4 The Method of Steepest Descent 3.5 The LMS Algorithm 3.6 The Learning Curve and Its Time Constants 3.7 Gradient and Weight-Vector Noise 3.8 Misadjustment Due to Gradient Noise 3.9 A Design Example: Choosing Number of Filter Weights for an Adaptive Predictor 3.10 The Efficiency of Adaptive Algorithms 3.11 Adaptive Noise Canceling: A Practical Application for Adaptive Filte 3.12 Summiiq Bibliography
Bibliography
5
Inverse Plant Modeling 5.0 Introduction 5.1 Inverses of Minimum-Phase Plants 5.2 Inverses of Nonminimum-PhasePlants 5.3 Model-Reference Inverses 5.4 Inverses of Plants with Disturbances 5.5 Effects of Modeling Signal Characteristics on the Inverse Solution 5.6 Inverse Modeling Error 5.7 Control System Error Due to Inverse Modeling Error 5.8 A Computer Simulation 5.9 Examples of Offline Inverse Modeling of Nonminimum-PhasePlants 5.10 Summary
6
Adaptive Inverse Control 6.0 Introduction 6.1 Analysis 6.2 Computer Simulation of an Adaptive Inverse Control System 6.3 Simulated Inverse Control Examples 6.4 Application to Real-Time Blood Pressure Control 6.5 Summary Bibliography
7
Other ConAgurations for Adaptive Inverse Control 7.0 Introduction 7.1 The Filtered-X LMS Algorithm 7.2 The Filtered-e LMS Algorithm 7.3 Analysis of Stability, Rate of Convergence, and Noise in the Weights f Filtered-€ LMS Algorithm 7.4 Simulation of an Adaptive Inverse Control System Based on the Filtered-e LMS Algorithm 7.5 Evaluation and Simulation of the Filtered-X LMS Algorithm 7.6 A Practical Example: Adaptive Inverse Control for Noise-Canceling Earphones 7.7 An Example of Filtered-X Inverse Control of a Minimum-Phase Plan 7.8 Some Problems in Doing Inverse Control with the Filtered-X LMS Algorithm
8.2 8.3 8.4 8.5 8.6 8.7 8.8
8.9 8.10 8.11 8.12 8.13 8.14 8.15 9
Proof of Optimality for the Adaptive Plant Disturbance Canceler Power of Uncanceled Plant Disturbance Offline Computation of Q t ( z ) Simultaneous Plant Modeling and Plant Disturbance Canceling Heuristic Analysis of Stability of a Plant Modeling and Disturbance Canceling System Analysis of Plant Modeling and Disturbance Canceling System Performance Computer Simulation of Plant Modeling and Disturbance Canceling System Application to Aircraft Vibrational Control Application to Earphone Noise Suppression Canceling Plant Disturbance for a Stabilized Minimum-Phase Plant Comments Regarding the Offline Process for Finding Q
System Integration 9.0 Introduction 9.1 Output Error and Speed of Convergence 9.2 Simulation of an Adaptive Inverse Control System 9.3 Simulation of Adaptive Inverse Control Systems for Minimum-Phase and Nonminimum-Phase Plants 9.4 Summary
10 Multiple-Input Multiple-Output (MIMO) Adaptive Inverse Control Systems 10.0 Introduction 10.1 Representation and Analysis of MIMO Systems 10.2 Adaptive Modeling of MIMO Systems 10.3 Adaptive Inverse Control for MIMO Systems 10.4 Plant Disturbance Canceling in MIMO Systems 10.5 System Integration for Control of the MIMO Plant 10.6 A MIMO Control and Signal Processing Example 10.7 Summary
11.6 Experiments with Adaptive Nonlinear Plant Modeling 11.7 Summary Bibliography
12 Pleasant Surprises A
Stability and Misadjustment of the LMS Adaptive Filter A. 1 Time Constants and Stability of the Mean of the Weight Vector A.2 Convergence of the Variance of the Weight Vector and Analysis of Misadjustment A.3 A Simplified Heuristic Derivation of Misadjustment and Stability Conditions Bibliography
B Comparative Analyses of Dither Modeling Schemes A, B, and C
Analysis of Scheme A Analysis of Scheme B Analysis of Scheme C A Simplified Heuristic Derivation of Misadjustment and Stability Conditions for Scheme C B.5 A Simulation of a Plant Modeling Process Based on Scheme C B.6 Summary Bibliography B. 1 B.2 B.3 B.4
C A Comparison of the Self-Thing Regulator of Astrom and Wittenmar with the Techniques of Adaptive Inverse Control C. 1 Designing a Self-Tuning Regulator to Behave like an Adaptive Inverse Control System C.2 Some Examples C.3 Summary Bibliography
D Adaptive Inverse Control for Unstable Linear SISO Plants D. 1 Dynamic Control of Stabilized Plant D.2 Adaptive Disturbance Canceling for the Stabilized Plant D.3 A Simulation Study of Plant Disturbance Canceling: An Unstable Pla with Stabilization Feedback D.4 Stabilization in Systems Having Both Discrete and Continuous Parts
Beam Control at the Stanford Linear Accelerator Center F. 1 Introduction F.2 A General Description of the Accelerator F.3 Trajectory Control F.4 Steering Feedback F.5 Addition of a MIMO Adaptive Noise Canceler to Fast Feedback F.6 Adaptive Calculation F.7 Experience on the Real Accelerator F.8 Acknowledgements Bibliography G Thirty Years of Adaptive Neural Networks: Perceptron, Madaline, and Backpropagation G. 1 Introduction G.2 Fundamental Concepts G.3 Adaptation -The Minimal Disturbance Principle G.4 Error Correction Rules - Single Threshold Element G.5 Error Correction Rules - Multi-Element Networks (3.6 Steepest-Descent Rules - Single Threshold Element G.7 Steepest-Descent Rules - Multi-Element Networks (3.8 Summary Bibliography H Neural Control Systems H. 1 A Nonlinear Adaptive Filter Based on Neural Networks H.2 A MIMO Nonlinear Adaptive Filter H.3 A Cascade of Linear Adaptive Filters H.4 A Cascade of Nonlinear Adaptive Filters H.5 Nonlinear Inverse Control Systems Based on Neural Networks H.6 The Truck Backer-Upper H.7 Applications to Steel Making H.8 Applications of Neural Networks in the Chemical Process Industry Bibliography
Glossary Index
design and behavioral characteristics are well known in the signal processing be used to control plant dynamics and to minimize the effects of plant disturb dynamic control and plant disturbance control are treated herein as two sep lems. Optimal least squares methods are developed for these problems, meth not interfere with each other. Thus, dynamic control and disturbance cancelling timized without one process compromising the other. Better control performan sult. This is not always the case with existing control techniques. Inverse control of plant dynamics involves feed-forward compensation, plant with a filter whose transfer function is the inverse of that of the plant. In pensation is well known in signal processing and communications. Every MODEM in the world uses adaptive filters for channel equalizati techniques are described here for plant dynamic control. Inverse control is fe control. The same precision of feedback that is obtained with existing control is also obtained with adaptive feed-forward control since feedback is incorpo adaptive algorithm for obtaining the parameters of the feed-forward compensa Inverse control can be used effectively with minimum phase and non-mini plants. It cannot work with unstable plants, however. They must first be stab conventional feedback, of any design that simply achieves stability. Then th stabilizing feedback can be treated as an equivalent stable plant that can be c the usual way with adaptive inverse control. Model reference control can be re porated into adaptive inverse control. Adaptive noise cancelling techniques are described that allow optimal r plant disturbance, in the least squares sense. Adaptive noise cancelling does n verse control of plant dynamics. Inverse control of plant dynamics does not a tive disturbance cancelling. If initial feedback is needed to provide plant stabil design of the stabilizer has no effect on the optimality of the adaptive distur celler. The designs of the adaptive inverse controller and of the adaptive disturbanc are quite simple once the control engineer gains a mastery of adaptive signal This book provides an introductory presentation of this subject with enough system design. The mathematics is simple and indeed the whole concept is easy to implement, especially when compared with the complexity of curr methods. Adaptive inverse control is not only simple, but it affords new control capa can often be superior to those of conventional systems. Many practical examp plications are shown in the text. Another feature of adaptive inverse control is that the same methods can be adaptive control of nonlinear plants. This is surprising because nonlinear pl have transfer functions. But approximate inverses are possible. Experimental r nonlinear plants have shown great promise. Optimality cannot be proven yet, bu
preparation of the drawings and final manuscript. BERNARD WIDROW Stanford, California EUGENE WALACH Haifa, Israel
Control Conce 1.0 INTRODUCTION
Adaptive filtering techniques have been successfully applied to adaptive antenna s [ 1-20]; to communications problems such as channel equalization [21-301 and ech celation in long-distance telephony [3 1-39]; to interference canceling [40-46]; to s estimation [47-571; to speech analysis and synthesis [58-60]; and to many other sign cessing problems. It is the purpose of this book to show how adaptive filtering algo can be used to achieve adaptive control of unknown and possibly time varying syste The system to be controlled, usually called the “plant,” may be noisy, that is, s to disturbances, and for the most part it may be unknown in character.’ The plant internal disturbances may be time variable in an unknown way. In some cases, the might even be unstable. Adaptive control systems for such plants would be advant over fixed systems since the parameters of adaptive systems can be adjusted or tailored unknown and varying requirements of the plant to be controlled. Adaptivity finds a n area of application in the control field [ 8 8 ] . In the past two decades or so, many hundreds of papers have been published on tive control systems in the Transactions of the IEEE Control Systems Society, in Au ica, in the IFAC (International Federation for Automatic Control) journals and conf proceedings, and elsewhere. At the same time, a very large number of papers on ad signal processing and adaptive array processing have appeared in the Transactions IEEE Signal Processing Society, Antennas and Propagation Society, Communicatio ciety, Circuits and Systems Society, Aerospace and Electronics Society, the Proce ofthe IEEE, and elsewhere. Many books have been published on these subjects. Th schools of thought, adaptive controls and adaptive signal processing, have develop most independently. The control theorists have by and large studied adaptive contro
‘Some prior knowledge of the character of the plant and its internal disturbances will be needed in establish proper control. For example, at least a rough idea of the transient response time of the plant wou quired in order to model it adaptively. Some idea of how rapidly the plant characteristics change for plants t over time would be needed. Some knowledge of the plant disturbance would be useful, such as disturbanc level at the plant output. Detailed knowledge of the plant and its disturbances would not be required how
the methodology of adaptive signal processing. The result is what we call “adap control.” We begin with a discussion of direct modeling (or identifying) the chara the unknown plant using simple adaptive filtering methods. Then we show h methods, with some modification but in a different configuration, can be used modeling (or equalization or deconvolution). Inverse plant models can be used plant dynamics. Next we show how both direct and inverse models can be used adaptive process to minimize the effects of plant disturbance. In this developm sume that the plant is completely controllable and observable, that it can (in a sense) be represented in terms of an input-output transfer function (albeit an unk and that the plant is stable (if unstable, someone has previously applied stabiliz back). The plant may be either minimum-phase or nonminimum-phase. The basic ideas of adaptive inverse control have been under development University over the course of many years. The earliest related work is described by Widrow on blood pressure regulation 1611. Subsequent work is reported in pers that were presented at Asilomar conferences (62,631. A Ph.D. dissertation Schaffer was concerned with model-reference adaptive inverse control [MI. A the work is given by Widrow and Stearns [65]. The first paper on adaptive inverse cluding adaptive plant disturbance canceling was presented by Widrow and Wal at the First IFAC Workshop in Control and Signal Processing in San Francisco [66 ond presentation was by Widrow in 1986 in a keynote talk at the Second IFAC on Adaptive Systems in Control and Signal Processing, University of Lund, Sw There have been almost no other publications on inverse control and disturbanc until recently. Several recent publications in the neural network literature hav concerning nonlinear adaptive inverse control [95,96,97].
INVERSE CONTROL
A conventional control system like the one illustrated in Fig. 1.1 uses feedback, response of the plant to be controlled, comparing this response to a desired res using the difference to excite an actuator or controller whose output drives the to cause the plant output to follow the desired response more closely. The system of Fig. I . 1 has unity feedback and is often called afollow-up sy the objective is that the plant output follow the input signal or the command i difference between the plant output and the command input signal is an error sig by the controller which amplifies and filters it to drive the plant to reduce the er
The use of feedback must be done in a careful way to prevent instability and satisfactory dynamic response. When the plant characteristics are time variable tionary, it is sometimes necessary to design the controller to vary with the plant. objective in doing this would be to minimize the mean square of the error. But ach objective is generally difficult. If one knew the plant characteristics versus time be able to determine the best controller versus time. Not knowing the plant, an id process could be used to estimate plant characteristics over time, and these cha could be used to determine the controller over time. Another idea would be to the controller and vary the parameters to directly minimize mean square error culty with this approach is that, regardless of how the controller is parametized square error versus the parameter values would be a function not having a unique and one that could easily become infinite if the controller parameters were push the brink of stability. The objective of the present work is to take an alternative look at the subje tive control. The approach to be developed, adaptive inverse control, in some sen open-loop control and it is quite different from the feedback-control approach We attempt to develop a form of adaptive control that is simple, robust, and pre some knowledge of the subject of adaptive filtering. adaptive inverse control is derstand and use in practice.
Command input
4 Controller
+
I
Figure 1.2
Basic concept of adaptive inverse control.
The basic idea of adaptive inverse control is to drive the plant with a sig controller whose transfer function is the inverse of that of the plant itself. The id trated with the system of Fig. 1.2. The objective of this system is to cause the plan follow the command input. Since the plant is generally unknown, it is necessary to adjust the parameters of the controller in order to create a true plant inverse. A
Comparing the system of Fig. 1.1 with that of Fig. 1.2, minimizing mean s is done in thefirst case by using the error signal directly in a feedback process f the plant input signal, whereas in the second case the error signal is used in process to control the parameters of the controller and is not fed back directly input. TheJirst case is feedback control and the second case is feedforward cont cases, feedback is used to ensure precise system responses. If adapting the controller of Fig. 1.2 were to make the error small, the contr have become an inverse of the plant. The cascade of the controller and plant wo have a combined transfer function matching a gain of unity. We assume that the plant is linear and that it varies slowly so that it is qua stationary. We assume that the controller has converged and that it too is line sistatically stationary. The dynamic characteristics of both the plant and contro represented by transfer functions, and the transfer function of the controller w reciprocal of that of the plant. If the plant has internal delay, the inverse controller may have difficulty in o it. The controller would need to be a predictor. Furthermore, if the plant is no phase (transfer function zeros in the right half of the s-plane or outside the unit z-plane), then the inverse controller would want to have poles in the right half of or outside the unit circle in the z-plane. Such an inverse would normally be unsta for overcoming these difficulties are developed in Chapter 5 . The system of Fig. 1.2 illustrates the basic concept of adaptive inverse co the adaptive algorithm works to control the parameters of the controller is a m developed below. Sometimes it is desired that the plant output track not the command inpu delayed or smoothed version of the command input. The system designer woul know the smoothing characteristic to be used. A smoothing model can be rea porated into the adaptive inverse control concept. How this may be done is il Fig. 1.3. The smoothing model is generally designated as the reference model in theory field. Thus, the system of Fig. 1.3 may be called a model-reference adap control system. The model-referenceidea is due to Whitaker, and an early refere A recent reference is [MI. The reference model is chosen to have the same dynamic response that t would like for the entire system. Referring to Fig. 1.3, it is evident that this resu obtained by once again adapting the controller to cause the mean square error to this case, the cascade of the controller and the plant after convergencewould hav response like that of the reference model. The product of the controller and pl functions would closely approximate the transfer function of the reference mod
Reference model
Figure 1.3
Model-reference adaptive inverse control.
Plant noise and disturbance present a problem for the adaptive inverse proach. Lack of feedback from the plant output back to the plant input perm plant noise and disturbance to exist unchecked at the plant output. Various sign ing methods for noise canceling have been developed [40-46], and with some m they have been applied to the cancelation of plant noise and disturbance. The ba is shown in Fig. 1.4. Use is made of both a plant model and a plant inverse model model has the same transfer function as the plant, while the plant inverse model fer function which is the reciprocal of that of the plant. How these models can b will be described below. For the present argument, we assume that these model Plant disturbance I
Command input
Sensor noise
Plant input
+
Plant mcdel
Nois disturb plant
Inverse Noise and disturbance filtered for cancelling
Figure 1.4
model Canceling plant noise and disturbance.
In the cont-cl literature, plant disturbance is often represented as an add at the plant input. Sensing the plant output is done with a detector or sensor th noisy. Sensor noise is often represented as an additive noise at the plant output. tem of Fig. 1.4, the plant noise and disturbance are separated from the plant’s dy put response. The plant input drives both the plant and its model (which is fre
uated at the plant output. One can show furthermore that the dynamic response is essentially unchanged even while the plant's noise and disturbance are canc disturbance canceling feedback. It is interesting to note that with perfect direct models, the system of Fig. 1.4 will be an unusual feedback system. The feedforw the major loop will have zero gain when the dynamics of the plant model perfect that of the plant itself. An adaptive inverse control system including the plant noise and disturba ing features of Fig. 1.4 and the model-reference control features of Fig. I .3 ar Fig. I .5. In a practical system of this type, separate adaptive processes would b obtain the plant model, the inverse plant model. and the controller. Plant di\turbance I
Sensor noise
Command input
-
+
Plant
v
Plant model algorithm Inverse plant model
4
Error
+
/
\ -
Reference model Figure 1.5
Model-reference adaptive invene control \y\teni wtth plant nnise and dislurhdnce
With regard to ordinary control systems, like the one shown in Fig. 1.1 plant noise and disturbance is done with feedback. But by incorporating feedba and disturbance control, the plant dynamics are inevitably altered. A comprom erally required in the design process to obtain good dynamic response and goo disturbance control all at the same time.
verify theory and to demonstrate workability.
SAMPLE APPLICATIONS OF ADAPTIVE INVERSE CONTROL
The objective of this section is to illustrate the application of adaptive inverse c variety of control problems. These problems are in some sense classic in natur will be discussed from many points of view in the chapters to follow. Here we s much explanation results of control by adaptive inverse control techniques appl of exemplary problems.
1.2.1 Dynamic Control of a Minimum-Phase Plant The plant to be controlled has the transfer function s
(s
+ 0.5
+ l)(s - 1 ) ’
This plant is minimum-phase and is unstable. The first step is to stabilize it with A root-locus diagram is shown in Fig. 1.6. It is clear from this diagram that th be stabilized by making use of the simple unity feedback system of Fig. 1.7, by loop gain within the stable range 00 > k > 2. The loop gain was set to k = 4 for t experiment. The closed loop transfer function is minimum-phase and has two p left half of the s-plane. The plant and its stabilization are continuous (analog) systems. The adapt control part, as it would be in the real world, is discrete (digital). A diagram of th system is shown in Fig. 1.8, including the necessary analog-to-digital conversion digital-to-analog conversion (DAC) components. The command input is sampled to both the inverse controller and the reference model. The controller output is to analog form, using a zero-order hold, to drive the plant and its stabilization error signal used to adapt the inverse controller is discrete. This is the differenc the reference model output and the sampled plant output. The system of Fig. 1.8 after that of Fig. I .3. Referring to Fig. 1.8, we define the discretized equivalent plant to be th transfer function from the digital input to the DAC, to the sampled output of the p includes the plant and its stabilization loop, shown in Fig. 1.7, and the DAC, an pler at the plant output. The impulse response of the discretized equivalent plan in Fig. 1.9. The chosen sampling rate was 10 samples per second.
'-05-1
-
-1 5 -2
Figure 1.6
+ '-
Figure 1.7
Root-locus of minimum-phare plant with proportional feedback.
k = 4
Plant
Minimum-phase plant stabilized with proportional feedback, k = 4.
The reference model chosen for this experiment was a one-pole digital f one-second time constant. Its impulse response is shown in Fig. 1.lo. The objec cause the overall system response, from the samples of the command input to the the plant output, to be as close to the response of the reference model as possible squares sense. The sampled command input was a first-order Markov process, g filtering white noise with a one-pole one-second time constant digital filter. Th was allowed to have 100 weights, and the theoretically optimal impulse respo many weights is shown in Fig. 1.11. The learned impulse response of the invers is shown in Fig. 1.12. Notice the similarity that it has to the optimal impulse re It is useful to compare the plant output to the reference model output. At ning, the inverse controller is learning and not yet performing well. The plant the reference model output are shown over the first 200 samples in Fig. 1.13. T track. It is too early in the process. The effects of learning can be observed in Fig. 1.14. The plant output beg properly after about 3,000 samples. This corresponds to about 300 seconds in rea entire sequence has 100,OOO samples, corresponding to 10,000seconds in real tim parison of the two outputs over the last 200 samples of the sequence is shown i
I
Figure 1.8
Model-r
Figure 1.9
Impulse response of discretized equivalent minimum-phase plant.
Time (0.1 sec.) Figure 1.10
Impulse response of reference model.
-0.05'' 0
10
20
30
40
50
60
70
80
90
Time (0.1 sec.) Figure 1.11
0.3-1 0.25 -
0.2-
3
-
0.15-
4
0.1
.-
Y
-
Impulse response of optimal 1 00-weight inverse controller.
-0.21 0
20
40
60
80
100
120
140
160
180
Time (0.1 sec.)
Plant output and reference model output during first 200 samples with minimum-
Figure 1.13
Tracking can be seen to be essentially perfect. Model-referenceinverse control d work in this case. 0.6
Plant Output Reference Model Output
I 04
0.2
._ -
c.)
8
0
-0.2
-0.4
-OBI
0
500
1,000
1,500
Time (0.1 sec.) Figure 1.14 learning.
2,000
2,500
J
3.0
Plant output and reference model output over sequence of 3,000samples, demonstra
-0.25b 0
Figure 1.15
1.2.2
20
40
80
80
100
120
140
180
180
Time (0.1 sec.)
Plant output and reference model output during last 200 samples of sequence of 100,
Dynamic Control of a Nonminimum-Phase Plant
For this experiment, a more challenging one, the plant to be controlled has the tra tion (S
(S
+
- 0.5) l)(S - 1)’
Since this plant is unstable, the first step is once again to stabilize it with feed plant cannot be stabilized with simple proportional feedback, like that of Fig. 1.7 be stabilized by using feedback with a compensating network. The compensati used in this experiment had the transfer function
A root-locus plot is shown in Fig. 1.16, and the stabilization feedback diagram i
Fig. 1.17. The compensating network has a zero at s = -1, and poles at s = The stable range for the loop gain is 00 > k > 20. For this experiment, k was ch k = 24. The result is a stabilized plant that remains nonminimum-phase.
1.2.3
Canceling Disturbance in the Minimum-Phase Plan
The stabilized plant was incorporated into an adaptive inverse control system li shown in Fig. 1.8. The sampling rate was chosen to be 10 samples per second. ence model was a one-pole digital filter with a one-second time constant, but i the reference model included an eight-second delay. The impulse response of the
-4
-
-6 -8
-
-10 -10
Figure 1.16
-8
-6
I
-4
-2
4
6
8
10
Root-locus plot for nonminirnum-phase plant with compensating netwo
(s
Figure 1.17
0 2 Real Axis
+ 7)(s - 2)
Nonminimum-phase plant with compensating network stabilization. k =
model including the delay is shown in Fig. 1.18. The delay is necessary for inv of a nonminimum-phaseplant. This will be explained in Chapter 5. The impulse response of the discretized stabilized plant, which includes th its stabilizer (shown in Fig. 1.17), and the DAC and the sampler at the plant shown in Fig. 1.19. This is the impulse response from the input to the DAC to output of the plant. The impulse response of the converged inverse controller is shown in Fig controller has 150 weights. It was obtained from a learning algorithm. The form of this impulse response could not be inferred intuitively. To observe and demonstrate workability, we compare the plant output wi ence model output when the entire system is driven by a command input. For iment, the command input was a first-order Markov process generated by app noise to a one-pole digital filter with a one-second time constant. Figure 1.21 of plant output and reference model output for the first 200 samples (the first 2 of a training sequence. Figure 1.22 shows 1,000 samples of this sequence. Lear observed with good tracking after about 500 samples, corresponding to 50 sec eration. The last 200 samples of the 20,000-sample sequence are shown in Fig
Time (0.1 sec.) Figure 1.18
Figure 1.19
Impulse response of reference model, including a delay of 8 seconds.
Impulse response of discretized stabilized nonminimum-phase plant with k
-0.05
-
I
-0.15 -0.1
I
-0.2
tracking is excellent, demonstrating precise control of a nonminimum-phase plan of adaptive inverse control, once learning has taken place.
0.30.25
-0.05
-
-
0
Figure 1.21
- Reference Model Output
20
40
60
80
100
120
140
180
180
Time (0.1 sec.)
Plant output and reference model output during first 200 samples with nonminimum
0
100
200
300
400
500
600
700
800
900
1,oo
Time (0.1 sec.)
Figure 1.22 Plant output and reference model output during sequence of 1,OOO samples, demonstra with a nonminimum-phase plant.
- Plant Output
____
Reference Model Output
-0 05 -0 1 -015-
-02’ 0
20
40
80
80
100
120
140
180
180
Time (0.1 sec.)
Figure 1.23 Plant output and reference model output dunng last 200 samples of training se nonminimum-phase plant.
An adaptive noise canceler fashioned after the one of Fig. 1.4 is used in this ex minimize the disturbance appearing at the plant output. The stabilized plant a canceler are integrated into a model-reference adaptive inverse control system diagrammed in Fig. 1.5. The resulting system is shown in Fig. 1.24. Referring to Fig. 1.24, an adaptive model was made of the equivalent pl converged impulse response is shown in Fig. 1.25. An adaptive inverse model o alent plant was made for use in the disturbance canceler, and its converged impul is shown in Fig. 1.26. The inverse controller was adapted, and its converged impulse response Fig. 1.27. It is useful to note that although this controller was adapted to con equivalent plant, and the learning process took place in the presence of plant d the converged dynamics of the controller were essentially unaffected by the p bance. The controller impulse response of Fig. 1.12 was adapted for the same plant, but without plant disturbance and without an adaptive disturbance cancele ison of Figs. 1.12 and 1.27 shows that both impulse responses are almost the separation of the functions of dynamic control and plant disturbance control is be discussed in detail in Chapters 8 and 9. This separation is a characteristic of t inverse control approach. The effectiveness of the adaptive disturbance canceler can be assessed spection of Fig. 1.28. The plant output disturbance (which is computed as the between the plant output and what the plant output would be if there were no d is squared at each sample time and plotted in Fig. 1.28. There is no averaging in the squared values are plotted over time. The disturbance canceling feedback loo until the five-thousandth sample in the time sequence. During this epoch, ther than enough time for adaptive modeling of the equivalent plant and adaptive in eling of this equivalent plant. Both models were required by the disturbance can the loop was closed and the immediate noise reduction became apparent. It wi in Chapter 8 that this form of disturbance canceler is optimal in the least squares when the plant is under dynamic control. The response of the overall system is exemplified by the plots of Figs. 1.29 ure 1.29 shows a comparison of the outputs of the reference model and the plant a ning of the learning sequence. The controller has not yet converged, and this ca error. Also, the disturbance canceler has not been turned on yet. Plant output contributes further to the overall error. Figure 1.30 shows the entire sequence o ror samples. The controller has learned its function after the first several hundr One can see the results of convergence of the controller at the beginning of the persists, however, because of the plant disturbance. The disturbance canceler i at 5,000 samples, and then the error is reduced to its lowest possible level. In Fi
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Model-reference inverse control system for minimum-phase plant with adaptive plant disturbance canceler.
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Figure 1.24
output
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Figure 1.28 Square of plant output disturbance versus time. The disturbance canceler was turned thousandth sample.
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Figure 1.29 Comparison of reference model output with plant output at the beginning of the learn Plant disturbance is present.
A N OUTLINE OR ROAD MAP FOR THIS BOOK
Section 1.1 has provided an overview of the idea of adaptive inverse control. Th treats dynamic control of the plant as a separate problem from that of control o turbance: 0
0
Dynamic control is effected to cause the overall system response from com signal to plant output signal to match that of a selected reference model.
Plant disturbance is controlled to minimize the mean square of the plant o and disturbance.
Treating these as two separate problems is a very effective approach since the p dynamic control and noise and disturbance control can be optimized without o mising the other. The basic block diagrams of Section 1.2 are highly simplified. There are m that need to be considered before one can successfully apply adaptive inverse
Figure 1.30 Entire learning sequence of 10,000 samples. Plot of error, equal to difference betw model output and plant output. Disturbance canceler turned on at 5,000 samples.
Figure 1.31 Square of overall system error plotted without averaging, over entire 10,000-sampl quence. Disturbance canceler turned on at 5,000 samples.
4
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Figure 132 Comparison of reference model output with plant output over last 200 samples of trai Controller has long ago converged. and plant disturbance IS being canceled.
practical problems. Our goal is to explain some of the many different techniq available for the practical application of adaptive inverse control. The theory various techniques is also derived and explained. This book contains 12 chapters and 8 appendices. Since we expect that will be new to most readers, we thought it appropriate to include in this introduct a “road map” of the book. We are addressing both signal processing engineers engineers. The methdology taught here, the mathematical techniques and the a nal processing techniques, will be more familiar to signal processing people. O hand, the problems that are being discussed are more familiar to control people We next describe the book and explain how it is organized.
Chapter 1: The Adaptive Inverse Control Concept
Chapter 1 introduces the idea of adaptive control and explains the need for it. It d more usual approaches to adaptive control and outlines the adaptive inverse cont its history. This chapter explains in an overall way how adaptive inverse control control plant dynamics, and separately, plant noise or plant disturbances. It brie the subject matter of each of the chapters of this book. The objective is to aid engineer and the signal processing engineer in gaining an understanding and a on adaptive inverse control and its applications.
Chapter 2: Wiener Filters
Wiener filters are best linear least squares filters which are very useful for predicti tion, interpolation, signal and noise filtering, and so on. To design them, prior kn
explained in detail This theory is extremely useful in the development of adapt control.
Chapter 3: Adaptive L M S Filters
This chapter reviews the theory of adaptive digital filtering, which is essential to opment of adaptive inverse control. This subject is discussed in many papers [ 1-6 them several by B. Widrow and co-authors. Adaptive filters are discussed at len eral textbooks such as Widrow and Steams [65], Haykin [69, 701 Cowan and G and Treichler and colleagues [72]. The chapter begins with the idea of an adaptive filter, a tapped delay line wi coefficients or tap weights driven by the LMS algorithm of Widrow and Hoff [73 is a gradient algorithm based on the method of steepest descent. The use of this to adjust the weights to minimize mean square error is described in the context practical applications. An important application is that of plant identification, ne adaptive control. Wiener theory is used to describe asymptotic adaptive behavior. of convergence and the effects of gradient noise (caused by obtaining gradients amounts of input-signal data) are analyzed. Fast adaptation causes noisy weight excess, can result in poor performance. The efficiency of adaptive algorithms is discussed. Two algorithms oper the same level of noise in their weights can be compared. The one that converg the more efficient. It is shown that when the input of an adaptive filter is station eigenvalues are equal or close in value, the LMS algorithm performs at a level of close to a theoretical maximum. When the eigenvalues are highly disparate, an similar to LMS but based on Newton’s method would be needed to approach the maximum in efficiency. For nonstationary inputs, the conventional steepest-de algorithm performs with close to maximum efficiency [77]. This chapter concludes with a description of adaptive filtering as applied t noise canceling. The principles of adaptive noise canceling are derived. Exper sults are presented illustrating the effectiveness of noise canceling techniques to in adult electrocardiography (removing 60-Hz interference) and to fetal electroca (removing interference from the maternal heart to reveal tiny signals from the fe
Chapter 4: Adaptive Modeling
The idea of LMS adaptive filtering for plant identification was discussed in Chap chapter analyzes the sources of error in adaptive plant modeling, such as the ef
ter 5 discusses inverse adaptive modeling. The direct model has a transfer func to that of the plant being modeled. The inverse model has a transfer function lik rocal of the plant transfer function. Forming a stable inverse transfer function is easy when the plant is minim But doing this for a nonminimum-phaseplant requires a two-sided Wiener impul which, to be causal, needs to be delayed and truncated. A theoretically optima obtained with the help of Shannon-Bode theory. The theory shows how appr verses can be formed for nonminimum-phase plants. Although such inverses a they can be very effective. Model-reference inverses can be formed in a similar way. The result is t cade of the plant and its inverse develops an impulse response equal to that o reference model. The inverse filter becomes the controller for the plant. The trol system then has a dynamic response like that of the reference model. Thi reference adaptive inverse control. If the plant being controlled is subject to disturbance, the plant inverse tained by adapting the inverse filter against a direct model of the plant instead of plant itself. Gradient noise in the weights of both the direct model and the inv affect the accuracy of the inverse model. Expressions are obtained for the vari error at the plant output due to noise in the inverse filter’s weights. This knowle important to the design of the adaptive controller.
Chapter 6: Adaptive Inverse Control
Schemes for adaptive inverse control were first presented in Chapter 1, but as sh they lack sufficient detail to be used in practice. Practical model-reference control systems are shown in Chapter 6 for b bance-free plants and for plants with enough disturbance to interfere with the in eling process. To alleviate the effects of plant disturbance, direct modeling is do the inverse modeling process is done offline using a disturbance-free plant mo the inverse. The idea of a dither signal, used in the direct modeling process, is here. If the dither is strong, good direct and inverse modeling is the result. Bu adds to the plant output disturbance. A theory is developed to optimize the po dither to minimize the summed effects of the control error and plant disturbance output. Adaptive control of blood pressure in experimental animals is reported. experiments were performed regulating a dog’s blood pressure in real time. T was in an induced state of shock. Under computer control, life-sustaining adm
to the algorithms of Chapter 6 are discussed in detail. Conditions for stability a Learning rates are obtained, as are expressions for weight noise variance. All of used to predict the performance of adaptive inverse control systems. A practical problem, canceling noise that leaks through earphones in a hig vironment, illustrates application of both the filtered-X algorithm and the filte rithm. This signal processing problem is very much like a control problem.
Chapter 8: Plant Disturbance Canceling
In the previous chapters a variety of techniques were developed for achieving namic control, even for plants subject to disturbance. These methods control plan but do nothing to control, reduce, or eliminate plant disturbance, however. An op tive scheme for reducing or eliminating plant disturbance is described and analy chapter. Its learning rate is determined. Shannon-Bode theory [68] is used to p mality and to derive expressions for the power of the residual plant output distur adaptive cancelation. It is proven for linear systems that no other method can mean square of the plant output disturbance to a lower level. Expressions are d overall system output mean square error for simultaneous application of adapt control and adaptive plant disturbance canceling. A simulation of an adaptive disturbance canceling system is reported. A of this system to real-time aircraft ride control is illustrated. Fast control of the ailerons could reduce the vertical component of disturbance due to random up downdrafts.
Chapter 9: System Integration
An entire control system consisting of an inverse controller, a plant to be control adaptive disturbance canceler for the plant is described and azalyzed in Chapte prising result develops from the analysis. If the plant model P ( z ) , which is use both the controller C ( z )and the noiz-canceling feedback filter Q(zhdoes not pe resent the plant P ( z ) , the errors in P ( z ) and the resulting errors in C ( z ) and Q(z in such a way that the overall ystem transfer function from commaEd input to p is unaffected by the errors in P ( z ) . In other words, small errorkin P ( z ) cause s in the design of the controller c^(z). In addition, small errors in P ( z ) cause small the transfer function of the plant P ( z ) in feedback connection with its disturbanc The effects of these two transfer function errors canceldeaving the overall syste function unaffected by small errors in the plant model P ( z ) .
the dynamic control of the plant. This is an important result. Although the pla controlled by an open-loop controller, feedback in the adaptive algorithms and among them result in precise dynamic control and close to optimal disturbance
Chapter 10: Multiple-Input Multiple-Output ( M I M O ) Ad Inverse Control Systems
MIMO systems are used when a physical plant to be controlled has multiple a of which have interacting effects on the process, and multiple sensors. Modeli trolling MIMO systems are much more complicated than doing the same with SI input single-output systems). This chapter begins with a review of digital signal processing for MIM The review section introduces notation used throughout the chapter. MIMO s be represented with block diagrams and flow graphs. Each signal line carries signals, that is, a signal vector. Transfer functions are matrices. The rules of ma apply, and transfer functions are not commutable. This has a considerable effec sign of adaptive algorithms to do plant modeling, inverse modeling, and plant canceling, all of which are described for MIMO systems. Two methods for devising MIMO inverse controllers are presented in C One is an algebraic technique, and the other is based on the filtered+ algorithm for adaptive plant disturbance canceling are also described. Integrated MIMO systems incorporating inverse dynamic control and p bance canceling in a single control system are shown. There are many ways to d eral of which are illustrated in Chapter 10. An application is described concerning the problem of reducing noise in a space by using several loudspeakers and several sensing microphones in a MIMO configuration. Control of noise in an airplane cabin is illustrated here as an exam are signal processing problems which are mathematically identical to control p
Chapter 11: Nonlinear Adaptive Inverse Control
The purpose of this chapter is to show how to do adaptive “inverse control” wit plants of both the SISO and MIMO types. Although nonlinear dynamic plants g not have inverses, techniques like inverse control can still be used. Many of the rules of MIMO systems apply to nonlinear systems, such a mutability of filtering operations. An additional rule for nonlinear systems is th havior should be modeled only with input signal character and power level cor to those of the actual plant input. Scaling and linearity do not work.
outgrowth of and a substantial generalization of the LMS algorithm. How to do nonlinear plant modeling, with and without dither, is described in ter. When using dither, the superposed natural plant command signals, which cou stationary and which sometimes could be larger or sometimes smaller than the nonlinearly in the plant. Simple direct modeling is not so simple. Means of de this interaction are explained. Once the plant model has been obtained, an inverse controller can be de ing this with the filtered-€ LMS algorithm is illustrated. Care is exercised to ens adaptive filters during training have input signals that have the right amplitude the right signal characteristics. Nonline? plant noise ceceling is demonstrated next. Online and offline for obtaining P, P-’,Q , and C are shown. The chapter ends with a block diag integrated nonlinear control system which could be SISO or MIMO. The system rates both plant disturbance canceling and nonlinear inverse dynamic control.
Chapter 12: Pleasant Surprises
This chapter summarizes the principal theoretical results of adaptive inverse c many of these results were unexpected and worked out so nicely that we called “Pleasant Surprises.”
Appendix A: Stability and Misadjustment of the LMS A Filter
This appendix summarizes learning theory for adaptive transversal filters based o adaptive algorithm. Key issues are learning rate, stability, and effects of noise in t (misadjustment). Most of this work has been published elsewhere by W colleagues [73-771. More precise stability conditions and formulas for misadju been reported by Horowitz and Senne [81]. Simplified derivations are given he vide substantially the same results as those of Horowitz and Senne. The work presented in this appendix is used throughout the book and is b gether here for the convenience of the reader.
Appendix B: Comparative Analyses of Dither Modeling S A, B, and C
Basically, there are four ways of doing plant modeling which are illustrated in The first method uses the natural plant input signals encountered during normal eration to do the modeling. The second method, scheme A, adds a dither signal to
of the adaptive modeling filter. Scheme C uses only dither for the input and for response of the adaptive plant model. All four of these methods are used throughout the book. They have adv disadvantages relative to each other. This appendix develops ranges of stable op ditions, learning rates, and misadjustment for all of the methods.
Appendix C: A Comparison of the Self-Tuning Regulator o and Wittenmark with the Techniques of Adaptive Inverse
Some of the finest work in adaptive control has been done by h o r n and Witten self-tuning regulator [82-881 is known worldwide for its simplicity,elegance, a can be used to control plant disturbance and plant dynamics, but it does not cont dependently like adaptive inverse control. The purpose of Appendix C is to com two approaches to adaptive control in order to appraise their relative strengths nesses. The self-tuning regulator can be represented as a system consisting of a p back controller, and a feedforward controller. The feedback controller can be minimize plant output noise and disturbance power. But the feedback changes ics. Compensation can be made for this, and at the same time, the overall resp made to match a model response M ( z ) by properly choosing the feedforward troller. It is shown in this appendix that an optimal feedback controller for a self ulator can always be chosen, and the resulting feedback portion of the system be stable. The requirements on the feedforward controller may turn out to make however. (This is unstable in the sense of having poles outside the unit circle in th When this happens, one can design a stable feedforward controller that can app realize the required transfer function with a delayed response. The self-tuning indeed a general methodology. Adaptive inverse control can always realize the optimal plant noise cance experiencing instability due to feedback. Furthermore, the ideal inverse contro stable, with its poles inside the unit circle if the plant is minimum-phase. Thi troller can be realized approximately by a transversal filter without delay. A sponse is only required when the plant is nonminimum-phase. Adaptive invers a general methodology. Adaptive inverse control and the self-tuningregulator are two completely d proaches to the problem of adaptive control of plant dynamics and plant disturb methods of adaptation differ, their adaptive filters are of different structure, the dynamic response are different, but their abilities to cancel plant disturbance
control. The reason for this is that a feedforward controller, even though it is ad leave the plant unstable. The first step in the utilization of adaptive inverse control for an unstable bilization with feedback. This appendix shows that the choice of stabilization not critical as long as the unstable plant is stabilized. The plant together with i stabilizer should be treated as a unit, as a stable equivalent plant. The equivale be outfitted with an adaptive disturbance canceler and with an inverse controll an ordinary stable plant. If two different feedback filters can stabilize the plant, it is shown in this ap the minimal plant output disturbance (after adaptive disturbance cancelation) for both stabilizing filters. Since no one stabilizer does better than any other, th stabilizer is not critical. All one needs is some form of feedback filter that will s plant, and the adaptive disturbance canceler will deliver optimal performance. The same is true for the design of the inverse controller. The choice of feed lization is immaterial. One can always design an inverse controller for the stabi If the inverse controller needs delay for its proper realization, the required delay pend on the choice of the stabilization filter. Also, the length of the inverse con not depend on this choice. The conclusion of Appendix D is that plant instability is no impediment to adaptive inverse control. One needs to know only enough about the plant to des back stabilizer for it. The design could be accomplished by experimentation. is not at all critical as long as the plant is stabilized. Once stabilized, the plant bilizer should be treated as a stable equivalent plant, and adaptive inverse con be applied in the usual way. Although a wide variety of stabilizer designs woul be acceptable, forming any one design would not always be a straightforward p indeed may require some effort, particularly if the unstable plant is nonlinear and
Appendix E: Orthogonalizing Adaptive Algorithms: RLS, DFT/LMS, and DCT/LMS
This appendix was prepared by Dr. FrancOise Beaufays, based on her Ph.D. diss search in the Department of Electrical Engineering at Stanford University. Extr value spread of the autocorrelation matrix of the input to an adaptive filter caus gence problems for the LMS algorithm. To speed up convergence and maintai she proposes a new adaptive algorithm, the DCTLMS. She begins with a brief discussion of the RLS (recursiveleast squares) algo algorithm is most often very effective, but it is complicated and has its own stab lems.
similar to DFT‘LMS, with the DCT (digital cosine transform) substituted for th Beaufays shows that the DCTLMS algorithm performs significantly better tha LMS algorithm when the adaptive filter input is first-order Markov, a type of s common in signal processing and controls. This type of signal comes from app noise to a one-pole filter. The DFT’LMS and DCTLMS algorithms are easy to implement and ar tionally robust. All of their parts, DFT or DCT, power normalization (like AG or TV), and LMS algorithm are “bulletproof.” These algorithms should enjoy ceptance and application in the future.
Appendix F: A MIMO Application: An Adaptive Noise C System Used for Beam Control at the Stanford Linear Acc Center
This appendix was prepared by Dr. Tom Himel, a research physicist at the St ear Accelerator Center (SLAC). It is based on his experience with control of th two-mile long linear accelerator on the Stanford campus whose output drives po electrons in opposite directions along the arcs of a circular collider. To achiev suitable for physics research, the electron and positron beams must be controlled to within several microns of each other after each travels distances of about thr Control of the beam is critical. The U.S. Department of Energy, the spon millions of dollars a year operating the accelerator and has spent many hundreds building it over the past 25 years. Physicists from all over the world depend on th for their research. Until recently, the beam was controlled by many servo loops positioned al manual control of set points. Intercoupling between stages has always been a p overall control. The beam output of one stage is the beam input for the next. Ad niques are now used to obtain precise local beam positioning and at the same tim decoupling between stages. The corrective methodology is based on adaptive noise canceling. The ad turbance canceler at each stage is an eight-input, eight-output MIMO system. The adaptive system (implemented in software) has been installed and w more than a year, 24 hours a day, 7 days a week. The result is much better accel ation. It is automatic, without the need of human operator intervention, and the fr collision events has increased. This is an operational system, no longer a laborat stration.
Appendix H: Neural Control Systems
This appendix shows how to construct nonlinear adaptive inverse control system ing use of neural networks and the backpropagation algorithm. It also describes a of neural control systems. One such application is the truck backer-upper. A simulated trailer truck by a neural controller while backing to a loading platform. After many backing many different initial conditions, the controller learns to steer the truck by ma mistakes and learning what not to do. Once learning is complete, the truck can b almost any initial state, states not previously encountered, and the controller stee while driving backward to the loading dock. The truck backer-upper is an exam a nonlinear controller can learn and, in a real sense, design itself. Principles of inverse neural control similar to those incorporated in the t ing system have been used to control the functioning of electric furnaces and fo process control. These applications are described in some detail and are not lab ercises. They are industrial applications of high commercial significance and ar of what is currently being called “intelligent control systems.” This completes the road map of the book. We now invite you to travel dow with us. We wish you a productive journey.
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2.0
INTRODUCTION
Wiener filters are best linear least squares filters which are used for prediction, estim interpolation, signal and noise filtering, and so forth. To design them, prior knowledg appropriate statistical properties of the input signal(s) is required. The problem is th prior knowledge is often not available. Adaptive filters are used instead, making use o data to learn the required statistics. Wiener filter theory is important to us however, b the adaptive filters used here converge asymptotically (in the mean) on Wiener sol Understanding Wiener filters is therefore necessary for the understandingof adaptive Wiener filter theory and adaptive filter theory are fundamental to adaptive inverse c The idea of best linear least squares filtering was introduced by Norbert Wi 1949 [ 11. The purpose of this chapter is to explain how Wiener filters work and ho can be designed, given the statisticalproperties of the input signals. Simple forms of W filters can be made optimal without regard to causality. More complicatedWiener filt be designed, using the Shannon-Bode [2] approach, to be causal and optimal in th squares sense. A nice perspective on this is given by Kailath [3]. The discussion of Wiener filters will be made with regard to discrete-time dig ters rather than analog filters. The reason for this is that the modern implementa Wiener filters and adaptive filters is digital almost everywhere. It should be noted ho that Wiener’s original work was analog, dealing with continuous rather than discrete s and systems. In the discussion to follow, Section 2.1 will describe correlation functions, their forms, and relations between input and output signals of linear discrete filters driv stochastic inputs. In Section 2.2, the two-sided (noncausal) Wiener filter will be de In Section 2.3, the Shannon-Bode approach to causal Wiener filter design will be giv
2.1
DIGITAL FILTERS, CORRELATION FUNCTIONS, Z-TRANSFORMS
Figure 2.1 shows a block diagram of a simple causal linear digital filter. The box beled “ z - ~ ” represent unit delays. The circles labeled “ho,” ‘%I,” and so on are w or gains, multiplying coefficients with no delay. The output of the filter is gk, whe
Figure 2.1
A causal linear digital filter.
the time index. The output is a weighted sum of the present input f k , and its d sions f k - 1 , f k - 2 , and so on. The filter of Fig. 2.1 is often called a tapped delay l transversal filter. The impulse response of the filter is a string of impulses whose h o , h 1 , and so on. The output gk can be expressed algebraically as
The above summation is a convolution of the input signal which can be represented symbolically as
The z-transform of the input
fk
fk
with the impulse re
is defined as
The z-transform of the output signal gk can be obtained from the definition in Eq the convolution in Eq. ( 2 . 2 ) . Accordingly,
Summing over 1 gives G ( z )= H ( z ) . F ( z ) .
Convolution in the time domain therefore corresponds to multiplication in th domain. The z-transform of the impulse response is H ( z ) , and this is called function of the filter. A digital filter having a two-sided (noncausal) impulse response is shown The branches labeled “z” represent unit time advances. The branches labeled “ sent unit time delays, as before. In a real-time sense, the advances cannot exist since they must be perfect predictors. The delays, of course, can exist. The filte cannot be physically built for real-time operation. The response of such a filte proximated, however, in a delayed and truncated form. The output signal gp of Fig. 2.2 can be expressed as a convolution of the the impulse response:
The derivation is similar to (2.1) and (2.2). The z-transform of this output si written as G ( z )= H ( z ) . F ( z ) .
The derivation is similar to (2.6) and (2.7). Assume now that the input signal f k is stochastic, stationary, and ergodic the autocorrelation function of f k as A $ f f ( m )= E [ f k
’
fk+rnl*
where the symbol E [ . ] represents expectation. We multiply the time sequence lagged by m time delays, then we average. Since time average and ensemble equal ( f k is ergodic by assumption) the autocorrelation function can be writte average:
‘The order of summation in (2.4) is interchangeable as long as the series expansion for F(z), g and the corresponding expansion for G(z) have a common region of absolute convergence in the z
Output, gk
Figure 2.2
A noncausal two-sided digital filter.
The autocorrelation function can be approximatelycalculated from real data usin making N large, but finite. Given a digital filter like that of Fig. 2.1 or Fig. 2.2, the crosscorrelation fu tween the input f k and the output gk is defined as A
@ f g ( m )=
E[fk
'
gk+ml
= @gf(-m).
Using (2.2), this can be expressed as
Since hl is fixed for all 1, and since f k is stochastic, the expectation can be writt
=hm
*@ff(m).
Therefore, the crosscorrelation between input and output of a linear digital filter volution of the input autocomelation function with the impulse response. The s is obtained for causal or noncausal filters.
m
/=o
m
p=o
Making use of (2.14) and (2.16), we obtain
Equation (2.17) is similar to a convolution. To place it in such ' m n , de,..ie a tim impulse response as -
-
A
A
hl = h-1, or h-1 = hr.
Accordingly, oo
+gg(m)= C h - , 4 f R ( m+ I ) /=O
This is a convolution, and can be written as @gg(m)
= h m *4fg(m) = h-m * @ j g ( m ) .
Making use of (2.14), (2.20) becomes @gg(m)
= h-m
* hm * @ j f ( m ) .
This is a double convolution. This derivation was done assuming that the impul was causal. The same result can be obtained with a noncausal or with a two-sid response. It is useful to have the z-transform of (2.21). It can be written as
Two-sided Wiener filters are easiest to design because their impulse responses strained. We begin with this design. Refer to Fig. 2.3. The digital filter has an input signal and it produces an output signal. Thi be a Wiener filter if its impulse response is chosen to minimize mean square error is defined as the difference between the filter output and the desired response:
When working with Wiener filters, the desired response signal generally conceptually. The statistical properties of the imagined desired response signal a tistical relationship of this signal to the filter input signal are assumed to be kno filter designer. The situation is quite different when dealing with adaptive filter desired response exists as an actual signal which must be available as an input time adaptive algorithm in order to achieve learning and adaptation. The Wiener not learn. Its design is fixed, based on a priori statistical knowledge. The impulse response of the Wiener filter is obtained by finding an exp mean square error and minimizing this with respect to the impulse response. Squ sides of (2.24)and using (2.8), we get
Input
fk
h; = ?
T
O
uDigital tfilter p
u
t
Desired response dk
Figure 2.3
The Wiener filter.
g
k
response,
We must set this derivative to zero for all values of j . The result is the Wien response, h i , determined by
/=-a,
This is the Wiener-Hopf equation, and it is in the form of a convolution
Taking z-transforms of both sides yields H * ( z ) .@f/(z) = @ f d ( z ) , or
The transfer function of the Wiener filter is H*(z) and is easily obtained from forms of the autocorrelation function of the input signal and the crosscorrelati between the input and the desired response. The Wiener impulse response can b by inverse z-transformation of H*(z). Since the mean square error is minimized by use of the Wiener filter, an for minimum MSE can be obtained by combining the Wiener-Hopf equation (2.2 general expression for MSE given by (2.26):
I=-00
I=-x
A very useful fact regarding Wiener filters, that the crosscorrelation betw ror and the input signal to each of the tap gains or weights is zero, can be demo follows. In general, this crosscorrelation is E[fk-mckl
= E [ f k c k + r n l = E [ f k ( d k + r n - gk+rn)l fkdk+rn
-
2
hlfk-/+mfk
/=-a2
1
dk = Sk Desired response
Figure 2.4
A Wiener noise filter.
c 00
=
-
W
f
f
h
- 0.
rc-00
If the impulse response is optimized in accord with the Wiener-Hopf equation, 00
E[fkEk+ml
= 4fd(m) r=-m
h;#ff(m - 1 ) = 0.
We will make use of this fact about Wiener filters in subsequent chapters. The following example illustrates how Wiener filter theory may be used Fig. 2.4. A signal s k is corrupted by additive noise n k . Let their sum f k be th Wiener filter. The purpose of the filter is to eliminate the noise and reproduce th best possible in the least squares sense. So the Wiener filter input is fk
= sk
+
gk
= fk
* h;,
nk.
The output of the Wiener filter is and the desired response is dk
= sk.
For the present example, we are given that the autocorrelation function for the s
and the autocorrelation function of the noise n k is 2 d n n ( m )= j a ( m ) ,
-4
-3
Figure 2.5
0
-I
-2
1
2
3
4
m
A geometric autocomelation function.
where S( ) is the Kronecker delta function. The noise is thus “white,” uncor time, with a mean square value of 2/3, and a mean value of zero. The signal i over time, has a mean square value of 10/27, and a mean value of zero. The sign are assumed to be uncorrelated with each other. The transfer function of the Wiener filter is given by (2.30). We need to ob and @ f f ( z to ) get this function. We can first find #fd(m)and # f f ( m ) ,and then them. Since the signal and noise are uncorrelated with each other,
The z-transform of this equation is @ff(Z)
= @ss(z)
+ @nn(z).
The z-transform of the noise autocorrelation is
2 =3’ The z-transform of the signal autocorrelation is @nn(d
To evaluate this z-transform, examine the geometric autocorrelation function Fig. 2.5, given by &,(rn) = A t i m 1 ,
where A is a scale factor, and t is a geometric ratio less than one. In accord with the definition of the z-transform (2.3),the transform of the lation function of Fig. 2.5 can be expressed as
2
m=-rn
+ r z - ’ + r2z-2 + r3z-3 + . ..] + A [ r z + r 2 z 2+ r 3 z 3+ . . . I .
[Arlml]~-m = A [1
Figure 2.6
Region ctt coniergence for geometric autocorrelation fuiiction
The two sums are geometric and converge for certain values of 2. The first series absolutely where
lrzl'l < 1, or
121 >
r.
The second series converges absolutely where 1 r
Irzl < 1, or IzI < -. Therefore, the z-transform (2.23)exists where 1
r
>
17 1 .
> r.
Under this condition,
-
A(l -r2) ( 1 -rz-I)(l- rz)'
This function has two poles in the z-plane, as shown in Fig. 2.6. The figure show of convergence of the transform. Making use of (2.47),the transform (2.42) can be evaluated as
Combining this with (2.41), we obtain
We have used the fact that the signal and noise are uncorrelated with each oth ingly, @fd(Z)
= @ss(z).
We can now get the transfer function of the Wiener filter,
Using (2.49) and (2.48),
This transform resembles (2.47). By inspection, we can inverse transform (2.5
2 (A)3
Ikl
h* k-16
, all k.
This is the Wiener impulse response. The minimum MSE can be computed using (2.31): m
I=-m
10 5 =24' The ratio of minimum MSE to the mean square of the desired response is
5/24 --
10127
- 56.25%.
was obtained by differentiating the MSE with respect to all of the impulses in t response and setting the resulting derivatives to zero. Our interest now focuse alization of causal Wiener filters. Their impulse responses are constrained to negative time. The optimal causal impulse response has zero response for neg and has zero derivatives of MSE with respect to impulse response for all times e greater than zero. Accordingly, the causal Wiener-Hopf equation is
h7causal = 0, j < 0.
This is not a simple convolution like the Wiener-Hopf equation for the unconstr and special methods will be needed to find useful solutions. The approach de Shannon and Bode [2] will be used. We begin with a simple case. Let the filter input be white with zero-mea variance, so that
4ff(rn)= 6 h ) . For this input, the causal Wiener-Hopf equations (2.58)-(2.59) become
CI"=,,h~a,,,4ff(j - I ) = h;c,,,l
= $fd(j)? j 2 0 hTc,,,,, = 0, j < 0
With the same white input, but without the causality constraint, the Wiener-Hop would be
C h ; 4 f f ( j - 1 ) = h; = 00
&fd(j),
ail j .
I=--bo
From this we may conclude that when the input to the Wiener filter is white, t solution with a causality constraint is the same as the optimal solution without except that with the causality constraint the impulse response is set to zero for neg With a white input, the causal solution is easy to obtain, and it is key to Shannonfind the unconstrained two-sided Wiener solution and lop off the noncausal part domain. But you don't do this if the input is not white because, in that case, (2.61) do not hold. Usually, the input to the Wiener filter is not white. Accordingly the first Shannon-Bode realization involves whitening the input signal. A whitening fi ways be designed to do this, using a priori knowledge of the input autocorrelatio or its z-transform.
(1 - az-')( 1 - a z ) ( 1 - btA2)(1 - bZ2) . . . @ f f ( z )= A ( I - a z - l ) ( I - az)(1- ~ z - 2 ) ( 1 - ~ z - 2 ) .. . * With no loss in generality,2 assume that all the parameters a, b, c , . . . , a, magnitudes less than one. If @ff(z) is factored as @ff(Z)
= @;f(z)
'
@>(z),
we have
All poles and zeros of @Tf(z) will be inside the unit circle in the z-plane. A zeros of @,(z) will be outside the unit circle in the z-plane. Furthermore,
@),(z) = @&(z-l), aZf(z-9= @ & ( z ) .
The whitening filter can now be designed. Let it have the transfer functio
To verify its whitening properties, let its input be fk with autocorrelation funct and let its output be i k with autocorrelation function $ ~ (i m ) . Using (2.22), the t the output autocorrelation function is
The autocorrelation function of the output is
4 i i ( m ) = S(m).
*The only w e not included involves zeros of @ f f ( z )exactly on the unit circle. This is a spec requires special treatment. These zeros are assumed to be somewhere off the unit circle, and they a on it by a limiting process.
Figure 2.7
Shannon-Rode realization of causal Wiener filter.
Therefore the output i s white with unit mean square. It is important to note that the whitening filter (2.69)is both causal and stabl are all inside the unit circle in the z-plane. Furthermore, using t zeros of ing filter does nothing irreversible to the signal. The inverse of the whitening transfer function equal to @TJ(z),which is also stable and causal because @TJ(z its poles inside the unit circle. The whitening filter can be readily utilized since stable, and invertible with a causal, stable filter. The Shannon-Bode realization is sketched in Fig. 2.7. The input signal output of the causal Wiener filter is &, The desired response is dk. The error is The input is first applied to a whitening prefilter, in accord with (2.69) nothing irreversible to the input signal, it merely simplifies the mathematics. sequent filtering operation does not benefit from the whitening filter, it could e immediately by inverse filtering. The subsequent filter is easily designed, since its input is white. We represen fer function by Y",, ( 2 ) and design it without regard to causality, then lop off the part of its impulse response Without regard to causality, the best subsequent filter would have the transf
@If(z)
=-@Jd(Z) @,(Z)
To obtain this result, use is made of the unconstrained Wiener solution (2.30), t account that the input is whitened by (2.69). The subsequent filter is made causal by inverse transforming (2.72)into th main and lopping off the noncausal part. The causal part is then z-transformed transfer function. This operation cannot be done with z-transforms alone. A sp tion has been devised to represent taking the causal part:
The first step is to factor @ f f ( z ) in accord with (2.65), (2.66), and (2.67). this example is given by (2.49): 20 - 6~ - 6 ~ - I
- (1 - jz)( 1 - fz-’1
Therefore,
( I - fz-‘) @ ; / = ( I - 7 zI - I ) ’ @fd(z)
--(1 - iz)
@ff - (1 - f z ) .
is given by (2.52) and (2.48). From this and (2.76) we obtain 5/18
The filter Y*(z) is generally two-sided in the time domain. It must be stable. (2.77) to be stable, its two terms must correspond to time domain components I
-I
az -+ (right-handed time function). 1 - fz-’ I
I
1 -Jz
--f
(left-handed time function).
Any other interpretation would cause instability. The time function correspon first term is sketched in Fig. 2.8(a). The time function for the second term is Fig. 2.8(b). The time function for the sum of these terms is shown in Fig. 2.8(c). part of the sum is sketched in Fig. 2.8(d). Taking the z-transform of this time fun 1 1 1 Yc:”a,(z) = - -z-I -z-2 . .. 3 6 12
+
+
+
Including the whitening filter, the transfer function of the causal Wiener filter i
... -4
-3
-2
0
-I
1
2
3
4
Time, k
(a)
response
-1 3 ,,
I
T 1-fz
-I
...
1 -
9
21
Time, k
Impulse response
1
3 ~,
-I
-1
...
1
:
9
I12
21
1 -
24
...
24
...
3
4
(C)
Impulse
-4
-3
-2
-1
0
I
2
Time. k
(d)
Figure 2.8
Correspondence between time functions and z-transforms for noise filtering exa
Figure 2.9
Causal Wiener filter for noise filtering example.
-
1
5
1 - +-I
*
This is quite a different result from (2.55),the transfer function of the unconstra filter. The impulse response corresponding to the transfer function (2.79) is Fig. 2.9. It is expected that the causal Wiener filter would not perform as well as strained noncausal Wiener filter. To verify this, we need an expression for min square error of the causal Wiener filter. The general equation for mean square error is (2.26). The minimum MSE o Wiener filter is obtained by substituting the causal Wiener-Hopf Eq. (2.58) into The result is 00
(E[6,21)min = @ d d ( O ) - C h k u d @ f d ( O . cpyJa,
/=O
Equation (2.80)can now be used to find the minimum MSE. For the abov
From (2.79),the causal Wiener impulse response is hhd
= A3( ' ) 5
k
, kz0.
Accordingly,
The ratio of the minimum MSE to the mean square of the desired response is 6/27 = 60%. 10/27
0
The unconstrained Wiener filter (two-sided) is determined by a convolu as the Wiener-Hopf equation:
c m
h;$ff ( j - I ) = $ f d ( j ) ,
all 1.
k-00
0
The transfer function of the two-sided Wiener filter is
0
The minimum MSE of this filter is
0
The causal Wiener filter is determined by the causal form of the Wienertion, which is not a simple convolution: M
0
The transfer function of the causal Wiener filter is
0
The minimum MSE for this filter is
0
It has been established that for any Wiener filter, causal or two-sided, th relation between the error Ek and the signal input to any of the tap gains fk-,,, is zero.
1981).
3.0
INTRODUCTION
The theory of adaptive filtering is fundamental to adaptive inverse control. Adaptive are used for plant modeling, for plant inverse modeling, and to do plant disturbanc celing. At every step of the way, adaptive filtering is present. It is important to th the adaptive filter as a building block, having an input signal, having an output sign having a special input signal called the “error” which is used in the learning proces building block can be combined with other building blocks to make adaptive inverse c systems. The purpose of this chapter is to present a brief overview of the theory of ad digital filtering. We will describe several applications for adaptive filters, and will d stability, rate of convergence, and effects of noise in the impulse response.’ We will relationships between speed of adaptation and performance of adaptive systems. In g faster adaptation leads to more noisy adaptive processes. When the input environmen adaptive system is statistically nonstationary, best performanceis obtained by a comp between fast adaptation (necessary to track variations in input statistics) and slow a tion (necessary to contain the noise in the adaptive process). A number of these issu be studied both analytically and by computer simulation. The context of this study w restricted to adaptive digital filters driven by the LMS adaptation algorithm of Widro Hoff [ I]-[5]. This algorithm and algorithms similar to it have been used for many y a wide variety of practical applications [6]. We are reviewing a statistical theory of adaptation. Stability and rate of conve are analyzed first; then gradient noise and its effects upon performance are assesse concept of “misadjustment” is defined and used to establish design criteria for a s problem, an adaptive predictor. Finally, we consider an application of adaptive filte adaptive noise canceling. The principles of adaptive noise canceling are derived and experimental results are presented. These include utilization of the noise canceling niques to improve results of adult and fetal electrocardiography.
‘Since the parameters of the adaptive filter are data dependent and time variable during adaptation, the a filter does not have an impulse response defined in the usual way. An instantaneous impulse response can be as the impulse response that would result if adaptation were suddenly stopped and the parameters were instantaneous values.
I I
I I
I I
Adaptive filter
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
------------Synchronous samplers
f(1)
I
dk (Desire respons
Unknown dynamic system to be modeled (continuous) Figure 3.1
(continuo Modeling an unknown system by a discrete adaptive filter.
A N ADAPTIVE FILTER
The form of adaptive filter to be considered here comprises a tapped delay lin weights (variable gains) whose input signals are the signals at the delay-line taps to add the weighted signals, and an adaptation process that automatically seeks impulse response by adjusting the weights. Figure 3.1 illustrates the adaptive would be used in modeling an unknown dynamic system. This filter is causa finite impulse response (FIR). In addition to the usual input signals, another input signal, the desired resp be supplied during the adaptation process to generate the error signal. In Fig. 3. input is applied to the adaptive filter as to the unknown system to be modeled. of the unknown system provides the desired response for the adaptive filter. I plications, considerable ingenuity may be required to obtain a suitable desired r some form of equivalent) for an adaptive process. The adaptive filter of Fig. 3.1 is a discrete-time or digital filter. The unkno to be modeled is a continuous-timeor analog filter. The inputs to the adaptive filt fore sampled versions of the input and output signals of the unknown system. T of the adaptive filter are adjusted by an automatic algorithm to minimize mean s Adjustability is indicated by the arrows through the weights. When the weigh and the error becomes small, the impulse response of the adaptive filter will clo
combiner of Fig. 3.2, a subsystem of the adaptive filter of Fig. 3.1, comprising i nificant part. This combinational system can be connected to the elements of a p antenna to make an adaptive antenna [7], or to a quantizer to form a single artific that is, an adaptive threshold element (Aduline [ 11, [8] or threshold logic unit (T use in neural networks and in adaptive logic and pattern-recognitionsystems. It used as the adaptive portion of certain learning control systems [ 101, [ 111; as a k of adaptive filters for channel equalization [ 121; for adaptive noise canceling adaptive systems identification [ 141-[23]. In Fig. 3.2, a set of n input signals is weighted and summed to form an ou The inputs occur simultaneously and discretely in time. The kth input signal ve
xk = [xlk, XZk, * ..
9
Xlkv..
*
9
xnkl
T
*
The set of weights is designated by the vector
WT = [ W I , w 2 , . . . , W I , .. . , w,].
For the present analysis, let the weights remain fixed. The kth output signal wil
Input signal Xk
Weights W
XIk
X2k
Xlt
dk Desired response
Figure 3.2
Adaptive linear combiner.
A
+
MSE = 6 = E [ c z ] = E[d;] - 2 E [ d k x l ] w WTEIXkXl]W = E[d;] - 2PTW WTRW,
+
where the crosscorrelation vector between the input signals and the desired resp fined as
and where the symmetric and positive definite (or positive semidefinite) input matrix R of the X-input signal is defined as
It may be observed from (3.6) that the mean square error (MSE) performance f quadratic function of the weights, a bowl-shaped surface; the adaptive process w tinuously adjusting the weights, seeking the bottom of the bowl. This may be ac by steepest descent methods [24],[25]discussed below.
T H E GRADIENT A N D T H E WIENER SOLUTION
The method of steepest descent uses gradients of the performance surface in seek imum. The gradient at any point on the performance surface may be obtained b tiating the MSE function, Eq.(3.6), with respect to the weight vector. The grad is
Set the gradient to zero to find the optimal weight vector W * : W * = R-IP.
This relation has similarity to (2.29), the Wiener-Hopf equation for unconstra whose impulse responses could extend infinitely in both directions over time two-sided IIR, infinite impulse responses. I t is not as easy to compare (3.11) with Wiener-Hopf equation for causal filters. The various forms of the Wiener-Hopf e though similar, are especially devised to meet the constraints, or lack thereof, i the Wiener impulse response. The minimum MSE for the finite impulse response case is obtained from (3.6): tfl,ifl =
Eld;] - P' W'.
Substituting (3.10) and (3.12) into (3.6) yields a useful formula for MSE: ( = (min
+ (W - W')TR(W - W*).
Define V as the difference between W and the Wiener solution W*:
v:
(W - W*).
Therefore,
6 = (,,,in
+ V'RV.
Differentiation of (3.15) yields another form for the gradient: V = 2RV.
The input correlation matrix, being symmetric and positive definite or positive se may be represented in normal form as R = QAQ-' = QAQT,
where Q is the eigenvector matrix, the orthonormal modal matrix of R, and A is i matrix of eigenvalues: An]. A=diag[Al, A2? . . . , A,,, e
.
.
,
Equation (3.15) may be reexpressed as
6 = tmin +V~QAQ-~V. Define a transformed version of V as A
V' = Q - ' V and V = QV'.
Accordingly. Eq. (3.15 ) may be put in normal form as
= ern," + V"AV'.
The method of steepest descent makes each change in the weight vector proport negative of the gradient vector: Wk+l = wk
+ p(-vk).
The scalar parameter p is a convergence factor that controls stability and rate of The gradient at the kth iteration is vk. Using (3.14), (3.16), (3.17), and (3.20) becomes V;+l - (I - 2pA)V; = 0.
This homogeneous vector difference equation is uncoupled. It has a simple ge lution in the primed coordinates [4]which can be expressed as V; = (I
- 2pA)'Vb,
where Vb is an initial condition V'0 -- Wb - W*'. For stability of (3.25), it is necessary that
'I-L '0,
ll~illax
where A,, is the largest eigenvalue of R. From (3.25), we see that transients in coordinates will be geometric; the geometric ratio for the pth coordinate is
r,, = (1 - 2pAp).
Note that the pth eigenvalue is Ap. An exponential envelope can be fitted to a geometric sequence. If the b time,is considered to be the iteration cycle, time constant t pcan be determined r p = exp
1 (-$) = 1 - -1 + - .... 2!tp' tp
The case of general interest is slow adaptation, that is, large tp.Therefore, r p = (1 - 2pAp) 2 1 -
1
7P
or 1
Figure 3.3
Feedback model of steepest descent.
Equation (3.3 1) gives the time constant of the pth mode for slow adaptation, wi The time constant is expressed in number of iteration cycles. Steepest descent can be regarded as a feedback process where the gradien role of vector error signal. The process, if stable, tends to bring the gradient to ure 3.3 shows a feedback model2 for a stationary quadratic MSE surface being s the method of steepest descent. The model is equivalent to the following set of
Notice that this model has an input not mentioned earlier, gradient noise N k , w because gradients estimated at each iteration cycle with finite amounts of input d perfect or noisy.
T H E L M S ALGORITHM
The LMS algorithm is an implementation of steepest descent using measured o gradients:
+p(-vk). A
WktI = Wk
+
The estimate of the true gradient is ? = V N k , equal to the true gradient plu noise. The error E k of the adaptive linear combiner of Fig. 3.2 is given by (3.4 gradient estimate may be obtained by squaring the single value of ~k and differ as if it were the mean square error:
2This has been called performance feedback [I], [26].
ponents of the gradient vector are obtained from a single data sample, without p of the weight vector. Each gradient vector is an instantaneous gradient. Sinc dient estimates are obtained without averaging, they contain large noise compo noise, however, becomes averaged and attenuated by the adaptive process, whi low-pass filter in this respect. It is important to note that for a fixed value of W, the LMS gradient estim ased. This can be demonstrated in the following way. Using (3.34) together wi get
From (3.9), the formula for the true gradient, we obtain V = -2(P - RW).
Therefore,
E[?] = V.
Although these gradient estimates are noisy, many small steps taken in the direc negative instantaneous gradients will, on average, go in the correct direction descent. Many convergence proofs for LMS algorithms have appeared in the lite the years [27], [28], [29], [30], [31]. Each one demonstrates convergence under of assumptions. Equation (3.27) turns out to be sufficient for convergence of vector in the mean, but not sufficient for convergence of the variance of the we A stronger condition for convergence of mean and variance is needed, namely,
1 tr R
->p>o.
This condition for LMS stability is derived and discussed in [32]. A brief de (3.39) is also included in Appendix A. The trace of R, equal to the sum of the m of all the signals going into the weights, would be known, at least approximately linear combiner is connected to the tapped delay line of an adaptive filter, the su squares equals the power level of the signal going into the adaptive filter multip number of weights. One would generally have some idea of the power level o going into a filter, making it easy to apply (3.39).
MSE t k is a function of iteration number k obtained by averaging over the ense eration k. Using (3.251, but assuming no noise in the weight vector, Eq. (3.40) beco ek
= (mi”
= emin
+
VbT(I- ~ / . L A ) ~-A~( I/ . L A ) ~ V ~ V,T(I- ~ / . L R ) ~-R~( I/ . L R ) ~ V O .
When the adaptive process is convergent, it is clear from (3.40) that
and that the geometric decay in t k going from 60 to emin will, for the pth mode, ometric ratio of t$ and a time constant 1
A 1
The result obtained by plotting MSE against number of iterations is called th curve. It has a number of modes equal to the number of distinct eigenvalues of noise in the weight vector, actual practice will show 6 k to be higher than indicated The learning curve shows the reduction of MSE resulting from repeated applica adaptation algorithm.
GRADIENT A N D WEIGHT-VECTOR NOISE
Gradient noise will affect the adaptive process both during initial transients and state. The latter condition is of particular interest here. Assume that the weight vector is close to the Wiener solution. Assume, as b xk and dk are stationary and ergodic and that xk is uncorrelated over time, that
The LMS algorithm uses an unbiased gradient estimate A
v = -26kXk
= vk - N k ,
where vk is the true gradient and N k is a zero-mean gradient estimation noise vect above. When Wk = W * ,the true gradient is zero. But the gradient would be according to (3.34), and this would be equal to pure gradient noise: Nk
=2EkXk.
When wk = W*, E [ E ~=].$m;nin. Accordingly, COV[Nk] = 4ijminR.
As long as wk 'u W*, we conclude that the gradient noise covariance is given app by (3.48) and that this noise is stationary and uncorrelated over time. This co based on (3.44) and (3.46) and on the Gaussian assumption. Projecting the gradient noise into the primed coordinates, N; = Q-'Nk. The covariance of N; becomes = E[N;NLT] = E[Q-'NkN:Q]
COV";]
= Q-'cov[N~]Q = 4tminQ-IRQ = 4ijminA.
Although the components of Nk are correlated with each other, those of N; a uncorrelated and can, therefore, be handled more easily. Gradient noise propagates and causes noise in the weight vector. Account dient noise, the LMS algorithm can be expressed conveniently in the primed as N;). W;+I = W; p(-V;) = W; p(-V;
+
h
+
+
This equation can be written in terms of V; in the following way: Vi+l = V;
+ p(-2AV; + N;).
Note once again that, since the components of N; are mutually uncorrelated and s is diagonalized, the components of noise in V; will also be mutually uncorrelat Near the minimum point of the error surface in steady state after adaptiv have died out, the mean of V; is zero, and the covariance of the weight-vector be obtained as follows. Postmultiplying both sides of (3.52) by their transposes expected values yields E[V;+,V;:I]
+
= E[(I - 2pA)V;ViT(1 - 2pA)] w2E[N;NLT] +puE[N;V;f(I - 2pA)I p E [ ( I - 2pA)V;Nkr].
+
It has been assumed that the input vector Xk is uncorrelated over time; the gra Nk is accordingly uncorrelated with the weight vector wk, and therefore N; and correlated. Equation (3.53) can thus be expressed as EIV;+IV&I] = (I - %pA)E[V;VLT](I- 2pA) +p2 E [N; NLT].
When the value of the adaptive constant p is chosen to be small (as is consis converged solution near the minimum point of the error surface),
/.LA<< I. Equation (3.57) thus becomes cov[v;] = /.L&inI. The covariance of Vk can now be expressed as follows: C O V [ V= ~ ] E[VkV:] = E[QV;V;TQ-i] = QCOV[V:]Q-’= /.L&,,~,,I.
From this we conclude that the components of the weight-vector noise are all o variance and are mutually uncorrelated. This derivation of the covariance depe assumptions made above and embodied in Eqs. (3.44), (3.46), (3.47). and (3.58). found by experience, however, that (3.60) closely approximates the exact covari weight-vector noise under a considerably wider range of conditions than these as imply. A derivation of bounds on the covariance based on fewer assumptions has by Kim and Davisson 1331.
M I S A D J U S T M E N T D U E T O GRADIENT NOISE
Random noise in the weight vector causes an excess MSE. If the weight vector free and converged so that WI. = W’, then the MSE would be Crnln. Howeve not occur in actual practice. Because of gradient noise, the weight vector will be on the average, will be misadjusted from its optimal setting. The weight vector Brownian motion about the bottom of the MSE bowl, causing the average MSE to than &,,in. An expression for MSE in terms of V; is given by (3.40), from which w expression for excess MSE due to weight vector noise: (excess MSE) = ViTAV; . The average excess MSE is an important quantity. It can be expressed as n
v= I
We define the misadjustment due to gradient noise as the dimensionless ratio of excess MSE to the minimum MSE, A
M=
average excess MSE tmin
For the LMS algorithm, under the conditions assumed above, M=ptrR.
This formula works well for small values of misadjustment, 25 percent or less, assumption Wk
2
w*
is satisfied. The misadjustment is a useful measure of the cost of adaptability. M = 10 percent means that the adaptive system has an MSE only 10 percent g emin.
It is useful to relate misadjustment to the speed of adaptation and the weights being adapted. Since tr R equals the sum of the eigenvalues,
CA,, = n
M = c1
wnAave
p= I
where Aaveis the average of the eigenvalues. From (3.43), I p = 1- ( l ) 4p
Tpm,
or
1 =4p
(L) . rpmu
ave
Substituting (3.69) into (3.68) yields
. M=E(-!-) 4 *pin- ave
For the special case where all eigenvalues are equal, the learning curve has on constant rmw,and the misadjustment is given by n M = - = p tr R. 4Tmu
tion with 10percent misadjustment can generally be achieved with an adaptive se equal to 10 times the memory time span of the adaptive transversal filter. Adap will cause more misadjustment. Adapting slower will result in less misadjustme
A DESIGN EXAMPLE: CHOOSING NUMBER OF FILTE WEIGHTS FOR AN ADAPTIVE PREDICTOR
Figure 3.4 is a block diagram of an adaptive predi~tor.~ Its adaptive filter conve layed input X k - A into the undelayed input xk as best possible in the least squares s adaptive filter weights are copied into an auxiliary filter having a tapped delay-lin identical to that of the adaptive filter and the input xk is applied without delay t iliary filter, the resulting output will be a prediction of the input, a best linear le estimate of X k + A (limited by finite filter length and misadjustment). A computer implementation of the adaptive predictor was made using a sim put signal X k obtained by bandpass filtering a white Gaussian signal and adding other independent white Gaussian signal. Prediction was one time sample into that is, A = 1, using an adaptive filter with five weights, all initially set to zero.
3This same predictor was described by Widrow in [4]; it has been used for data compression an coding [341 and for “maximum entropy” spectral estimation [35], [36]. Adaptive filter
A W
Input Time delay
I
,/‘
copy weights
,I
EA-A
, I
I1
, /
1 b
W
b ik+A
Figure3.4 Anadaptive predictor. From B . WIDROW, J.M. MCCOOL,M.G. LARIMORE,and C.R Jr., “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” P roc. IEEE @, (August 1976), pp. 1151-1 162.
noiseless learning curve I
loo
200
300
Number of Adaptations
400
500
Learning curves for adaptive predictor. From B . WIDROW, J.M.MCCOOL,M.G.LA C.R. JOHNSON,Jr., “Stationary and nonstationary learning characteristics of the LMS adaptive filte 0,Vol. 64, NO.8 (August 1976), pp. 11.51-1 162.
Figure35
Figure 3.5 depicts three learning curves. For each adaptive step, the valu responding to the current weight vector wk was calculated from (3.13) using kn of R and !&,in, giving the individual learning curve. The smooth ensemble avera curve is simply the average of 200 such individual curves, and it approximates t behavior in the mean. The third curve calculated from (3.41) shows how the pro evolve if perfect knowledge of the gradient were available at each step. It is steepest-descent learning curve. Of particular interest is the residual difference, after convergence, between ble average learning curve and the noiseless steepest-descent learning curve. T course, converges to tmin. The difference is the excess MSE due to gradient n case, giving a measured misadjustment of 3 percent. The theoretical misadju M = 2.5 percent. The minor discrepancy was due mainly to the fact that the in were highly correlated, in violation of the assumption that E[XkX&,] = 0, V in the derivation of misadjustment formula (3.7 1). The ensemble average learning curve had an effective measured time con about 50 iterations since it fell to within 2 percent of its converged value at arou 200. When all eigenvalues are equal, Eq. (3.43) becomes 1
n
r,, = - = 4pA 4ptrR’
Using (3.72) in the present case (although the eigenvalues ranged over a 10 to obtained r,, = 50, which agreed with the experiment. Equation (3.72) gives a an effective time constant, useful even when the eigenvalues are disparate. The performance of the adaptive filter may have improved with an incr number of weights. However, for a fixed rate of convergence, larger numbers increase misadjustment. Figure 3.6 shows these conflicting effects. The lowes
5
15
10
20
Number of Weights n
Figure 3.6 Performance versus number of weights and adaptive predictor time constant. From el. al., “Stationary and nonstationary learning characteristics of the LMS adaptive filter.” Proc. lEE No. 8 (August 1976). pp. 1151-1 162
r,, = 00, represents idealized noise-free adaptation providing the minimum MS for each value of n. The other curves include average excess MSE due to gradien define the average M S E to be the sum of the minimum MSE and the average ex Thus (average MSE) = 1 1 M ] { m , n ( n ) .
+
emln(n)g o e s down with increasing n, but M goes up in proportion t o n . Using th
theoretical curves have been plotted in Fig. 3.6 for approximate values of r, 25, and 15 iterations. It is apparent from these curves that increasing the number does not always guarantee improved system performance. Experimental points computer simulation have compared very well with theoretical values predicted Typical results are summarized in Table 3. I . TABLE 3.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL ADAPTIVE PREDICTOR PERFORMANCE
Number of Weight\
Approximate Time Constant
n
‘mu
-
5
5 5 5 10 10
100 50 25 15 100 50
Average MSE
Misadjustment
Theoretical
Exprirnental
Theoretical
0 742
0 751 0 754
13% 25 50 83 25 50
0751 0 769 0 794 0 737 0 755
0 781 0 824 0 745 0 764
Expen
2
3 6
I2 3 6
Appendix A presents alternative derivations of stability bounds for LMS a adjustment. These derivations are more rigorous than those presented above. F
ment versus rate of adaptation. The question arises: Could another algorithm that would produce less misadjustment for the same rate of adaptation? Suppose that an adaptive linear combiner is fed N independent input d XI, X2, . . . , Xk, . . . , XN drawn from a stationary ergodic process. Associate of these input vectors are their scalar desired responses dl, d2, . .. , d,v,respec drawn from a stationary ergodic process. Keeping the weights fixed. a set of N tions can be written as c?k = dk - W T & . k = 1, 2, . . . , N.
Let the objective be to find a weight vector that minimizes the sum of the squ error values based on a sample of N items of data. Equation (3.74) can be written in matrix form for all of the error values a &=D-XW,
where X is an N x n rectangular matrix
X
A
=
[XIX2...XN]T,
where E is an N-element error vector A r &= [€,62...CNI,
and where D is an N-element desired-response vector
D = [dld2...d,qIr. A
A unique solution for the weight vector which brings E to zero exists only if X and nonsingular. However, the case of greatest interest is that of N >> n. The squares of the errors is
E T & = DTD + W T X T X W- 2 D T X W . This sum when multiplied by I/N is an estimate 8 of the MSE, 6. Thus 1 6 = - E T E and lim 6 = 6 .
N
N-m
Note that 6 is a quadratic function of the weights. The parameters of the qua (3.78) are related to properties of the N data samples. ( E T E ) is square and po nite or positive semidefinite. $ is the small-sample-size MSE function, while 6 i sample-size MSE function. These functions are sketched in Fig. 3.7. The function $ is minimized by setting its gradient to zero. We will use symbol for a gradient operator. Accordingly:
V< = 2XTXW - 2XTD.
W’
Figure 3.7 Small- and large-sample size MSE curves. From B . WIDROW et. al., “Stationary and learning characteristics of the LMS adaptive filter,” Proc. IEEE 0,Vol. 64, No. 8 (August 1976). p
The “optimal” weight vector based only on the N data samples is
6’* = (XTX)-’XTD.
To obtain this result, it is necessary that (X’X)be positive definite. Equation small-sample-size Wiener-Hopf equation. This formula gives the position of the of the small-sample-size bowl. The corresponding formula for the large-sampl is the Wiener-Hopf equation (3.10). Making use of the N data samples, we could calculate %* by making us by a training process such as LMS, or by some other least squares optimization Taking the firstklock of N data samples, we obtain a small-sample-size functio minimum is at W;. Th& could be repeated with a second data sam_ple, giving a whose minimum is at W;, and so on. Typically, all the values of W* would diff true optimum W* and would, thereby, be misadjusted. To analyze the misadjustment, assume that N is large and that the typical curve approximately matches the large-sample-size curve. Therefore,
( % 6 and (6 - ()
d6.
The true large-sample-size function is
6 = &,ni,
+ V‘’AV’.
The gradient of this function expressed in the primed coordinates is V’ = 2hV’. A differential deviation in the gradient is
(dV’) = 2A(dV’)
+ 2(dA)V’.
This deviation could represent the difference in gradients between small- and lar size curves.
Equation (3.50) indicates that when the gradient is taken from a single sample of the covariance of the gradient noise when W‘ = W*’is given by 44,,,inA. If the gr estimated under the same conditions but using N independenterror samples, this would be 4 C O V [ ~ V=’ ]-&,inA. N Substituting (3.88) into (3.87) yields 1 COV[~V’] = -tminA-’. N The average excess MSE, an ensemble average, is (average excess MSE) = E[(dV’)TA(dV’)]. Equation (3.89) shows cov[dV’] to be diagonal, so that
n (average excess MSE) = $,,,inin.
Normalizing this with respect to tmin gives the misadjustment: n (number of weights) M=-= N (number of independent training samples) *
This formula was first presented by Widrow and Hoff [ 11 in 1960. It ha for many years in pattern recognition studies. For small values of M (less than 2 it has proven to be very useful. A formula similar to (3.92), based on somewh assumptions, was derived by Davisson [37] in 1970. Although Eq. (3.92) has been derived for training with finite blocks of da used to assess the efficiency of steady-flow algorithms. Consider an adaptive filter with stationary stochastic inputs, adapted by the LMS algorithm. For sim all eigenvalues of the input correlation matrix R be equal. As such, from (3.71 n M=-. 4 rm, The LMS algorithm exponentially weights its input data over time in determin weight values. If an equivalent uniform averaging window is assumed equal to t settling time, approximately four time constants, the equivalent data sample t instant by LMS is essentially Neq = 4rm, samples. Accordingly, for the LMS a
n
M=--
Neq
-
(number of weights) (number of independent training samples) ‘
or h
wk+I = wk
-k 2/LR-’€kXk.
This algorithm is a steady-state form of recursive least squares (RLS). The ma recursively updatcc! from the input data [40]. The process causes all adaptive mo essentially the same time constant. Algorithms based on this principle are poten efficient than LMS but are more difficult to implement. RLS algorithms are descr in Appendix E.
ADAPTIVE NOISE CANCELING: A PRACTICAL A PPLICAT I0N FOR A DA P T IV E FILTER ING
Many practical applications exist for adaptive FIR filters of the type described in ter. Among the more interesting of these applications is adaptive noise cancelin is to subtract noise from a noisy signal to obtain a signal with greatly reduced n done properly with adaptive filtering techniques, substantial improvements in noise ratio can be achieved. Separating a signal from additive noise, even when their respective pow overlap, is a common problem in signal processing. Figure 3.8(a) suggests a c proach to this problem using optimal Wiener or Kalman filtering [42]. The pur optimal filter is to pass the signal s without distortion while stopping the noise eral, this cannot be done perfectly. Even with the best filter, the signal is distorted noise does go through to the output. Figure 3.8(b) shows another approach to the problem, using adaptive noise A primary input contains the signal of interest s, and an additive noise n. A nois input is assumed to be available containing n I , which is correlated with the o rupting noise no. In Fig. 3.8(b), the adaptive filter receives the reference nois and subtracts the result from the primary input. From the point of view of the a ter, the primary input ( s no) acts as its desired response and the system outpu error. The noise canceler output is obtained by subtracting the filtered reference the primary input. Adaptive noise canceling generally performs better than the c proach since the noise is subtracted out rather than filtered out. In order to do ada
+
40ther algorithms based on LMS have been devised to give fast convergence and low misadju employ a variable p [38]. Initial values of p are chosen high for rapid convergence; final values of low for small misadjustment. This works well as long as input statistics are stationary. This proc methods of stochastic approximation on which it is based will not perform well in the nonstationary
I I
I
I
I I I
,
Reference I input I
I
I I
Error E
---------------
1 J
Adaptive noise canceler (b)
Figure 3.8 Separation of signal and noise: (a) classical approach; (b) adaptive noise-canceling ap B . WIDROWet at., “Adaptive noise canceling: Principles and applications,” Pmc. lEEE @, Vol. 6 cember 1975). pp. 1692-1716
canceling, one might expect that some prior knowledge of the signal s or of th and n I would be needed by the filter before it could adapt to produce the nois signal y. A simple argument will show, however, that little or no prior knowled or n 1 or of their interrelationships is required, except that n 1 is correlated with s is uncorrelated with both no and nl . Assume that s, no, n 1 , and y are statistically stationary and have zero mea that s is uncorrelated with no and n I and suppose that n I is correlated with no. is €=S+no-y. Squaring, one obtains E2
= s2
+ (no - y)2 + 2s(no - y).
Taking expectations of both sides of Eq. (3.98) and realizing that s is uncorrela and with y , yields
+ +
E k 2 I = E[s21 El(n0 - y)21 + 2E[s(no- Y)l = Eb21 E[(no- yl21.
Adapting the filter to minimize E [ c 2 ]will not affect the signal power E [ s 2 ] .A the minimum output power is (E[€21),i, =
+
(“0
- Y)21)mi”.
and adjustability of the adaptive filter and for the given reference input. The ou contain the signal s plus noise. From (3.101) the output noise is given by (no minimizing E [ r 2 ]minimizes E [ (no - Y ) ~ ]minimizing , the total outputpower mi output noise power. Since the signal in the output remains constant, minimizin output power maximizes the output signal-to-noise ratio. Adaptive noise canceling has proven to be a very powerful technique for cessing. The first application was made to the problem of canceling unwanted 50-Hz) interference in electrocardiography [43]. The causes of such interferenc netic induction and displacement currents from the power line, and ground loop tional filtering has been used to combat 60-Hz interference, but the best approac be adaptive noise canceling. Figure 3.9 shows the application of adaptive noise canceling in electrocar The primary input is taken from the ECG preamplifier; the 60-Hz reference inp from a wall outlet. The adaptive filter contains two variable weights, one applied t ence input directly and the other to a version of it shifted in phase by 90". The tw versions of the reference are summed to form the filter's output, which is subtr the primary inpu:. Selected combinations of the values of the weights allow th waveform to be changed in magnitude and phase in any way required for opti lation. Two variable weights, or two degrees offreedom, are required to cance pure sinusoid. A typical result of an experiment performed by digital processing is sho 3.10. Sample resolution was 10 bits, and sampling rate was 1,OOO Hz. Figure 3.1 the primary input, an electrocardiographic waveform with an excessive amoun interference, and Fig. 3.10(b) shows the reference input from the wall outlet. Fig is the noise canceler output. Note the absence of interference and the clarity of the adaptive process has converged. Another useful application of adaptive noise canceling is one that involve interference from the mother's heart when attempting to record clear fetal ele grams [43]. Figure 3.11 shows the location of the fetal and maternal hearts and ment of the input leads. The abdominal leads provide the primary input (conta ECG and interfering maternal ECG signals), and the chest leads provide multipl inputs (containing pure interference, that is, the maternal ECG). Figure 3.12 show figuration of the adaptive noise canceler. This is a multiple-reference noise c works much like a single-channel canceler. Figure 3.13 shows the results. Th ECG from the chest leads was adaptively filtered and subtracted from the abdom leaving the fetal ECG. This was an interesting problem since the fetal and mat signals had spectral overlap. The two hearts were electrically isolated and worke dently, but the second harmonic frequency of the maternal ECG was close to the
Adaptive noise canceler I
I
I
To 60-Hz wall outlet
I
+
I I
rn input
I
,
I I
90”Delay
I
r
II )I
r
I
60-Hz interference
Figure 3.9
I I I
Primary input
Canceling 60-Hz interference in electrocardiography. From B . WIDROWet. al., “ canceling: Principles and applications,” Proc. IEEE 0.Vol. 63, No. 12 (December 1975). pp. 169
Figure 3.10
Result of electrocardiographic noise-canceling experiment. (a) primary input; (b) re B. WlDROW ET. A L . "Adaptive noise canceling: Principles and Pm./ E E E @ . Vol. 63, No. 12 (December 1975). pp. 1692-1716. (c) noise canceler output. From
Mother's cardiac vector
Chest leads Neuual electrode Abdominal led placements
Fetal cardiac vector
(a)
Figure 3.11 Canceling maternal heartbeat in fetal electrocardiography, (a) cardiac electric fi mother and fetus. (b) placement o f leads. From B. W I D R O W ET A L ... "Adaptive noise canceling: applications." Proc. /,FEE@. Vol. 63. N o . 12 (December 1975). pp. 1692-1716.
tal of the fetal ECG. Ordinary filtering techniques would have had great difficul problem.
SUMMARY
This chapter described the performance characteristics of the LMS adaptive filt filter composed of a tapped delay line and adjustable weights, whose impulse controlled by an adaptive algorithm. For stationary stochastic inputs, the mean s
2 Reference inputs
’
3
4
Multiple-reference noise canceler used in fetal ECG experiment. From B. WID “Adaptive noise canceling: Principles and applications,” Proc. IEEE @, Vol. 63, No. 12 (De pp. 1692-1716.
Figure 3.12
(C)
Figure 3.13 Result of fetal ECG experiment (bandwidth, 3-35 Hz; sampling rate, 256 Hz): (a) r (chest lead); (b) primary input (abdominal lead); (c) noise canceler output. From B . WIDROW ET. A noise canceling: Principles and applications,” Proc. IEEE 0,Vol. 63, No. I2 (December 1975). p
and speed of adaptation when the eigenvalues of the input correlation matrix a close in value. When the eigenvalues are highly disparate ()cmax/A.min > lo), an similar to LMS but based on Newton's method would approach this theoretica closely. For an adaptive linear combiner or an adaptive FIR filter, the Wiener weig
W* = R-'P. The mean square error is
+ V'RV.
6 =ni,-& The minimum MSE is fmin =
E [ d 3 - P'W*.
When adapting with the method of steepest descent, the time constant for the p the weights as they relax toward the Wiener solution is 1
rp = 2p).p, small 1-1.
The time constant for the pth mode of the mean square error as it relaxes toward small p .
The LMS algorithm is based on the method of steepest descent and it uses ins gradient estimates based on single samples of data. LMS is given by
wk+I
= Wk
+ 2pEkXk.
The condition on the convergence parameter p for stability of the variance of vector when adapting with LMS is 1 tr R
-->p>o.
When adapting with the LMS algorithm, the misadjustment is
M = ptrR.
Processes involving adaptation with a finite number N of data samples have a sum of errors given by E'E = D'D
+ WTXTXW- 2DTXW.
n M = - = N
(number of weights) (number of independent training samples) *
This formula applies when the weight vector is chosen to be optimal for the giv data sample. The weight vector goes to the bottom of the small-sample-size MS Since MSE is minimized for the finite data sample, the data are used with ma ciency. The formula for misadjustment of an FIR filter in steady flow with the tation is very similar to (3.92) when the eigenvalues are equal. This suggests t as efficient in its use of data as an algorithm can be when the eigenvalues are e proximately so. With highly disparate eigenvalues, an LMS algorithm based o method could be the most efficient. Such an algorithm would be
+ 2/LR-‘€kXk. h
Wk+] = wk
This is a form of recursive least squares, RLS. Good methods for calculating R from the input data are given in [41] and are discussed in Appendix E.
Bibliography for Chapter 3
“Adaptive switching circuits,” in IRE [ l ] B. WIDROW,and M.E. HOFF, Conv. Rec., 1960, Pt. 4, pp. 96-104.
[2] B. WIDROW, and J.M. MCCOOL,“A comparison of adaptive algorithm the methods of steepest descent and random search,” IEEE Trans. Antenn Vol. AP-24, No. 5 (September 1976), pp. 615-637.
J.M. MCCOOL,M.G. LARIMORE, and C.R. JOHNSON, JR. [3] B. WIDROW, ary and nonstationary learning characteristics of the LMS adaptive filter,” P Vol. 64,N0.8(AuguSt 1976),pp. 1151-1162.
[4] B. WIDROW,“Adaptive filters,” in Aspects of network and syste ed. R.E. Kalman and N. De Claris (New York: Holt, Rinehart and 1970), pp. 563-587.
and E. WALACH, “On the statistical efficiency of the LMS 151 B. WIDROW, with nonstationary inputs,” IEEE Trans. Info. Theory -Special Issue on Ad tering, Vol. 30, No. 2, Pt. 1 (March 1984), pp. 211-221. [6] See references [ 1-58] of Chapter 1.
tom. Control, Vol. AC- 1 1 (January 1966), pp. 71-77.
[ 1 I ] B. WIDROW,“Adaptive model control applied to real-time blood-press
tion,” in Pattern Recognition andMachine Learning, Proc. Japan-US. Sem Learning Process in Control Systems, ed. K.S. Fu (New York: Plenum Pr pp. 3 10-324. [ 121 See references 12 1-30] of Chapter I. [ 131 See references [ 4 0 4 6 ] of Chapter 1.
[ 141 P.E. MANTEY,“Convergent automatic-synthesis procedures for sample
works with feedback,” TR No. 7663- 1, Stanford Electronics Laboratories CA, October 1964.
[ 151 P.M. LION,“Rapid identification of linear and nonlinear systems,” in P
JACC, Seattle, WA, August 1966, pp. 605-615, also A I M Journal, Vol. 1967), pp. 1835-1842. [ 161 R.E. Ross and G.M. LANCE,“An approximate steepest descent method
eter identification,” in Proc. 1969 JACC, Boulder, CO, August 1969, pp. 4
[ 171 R. HASTINGS-JAMES, and M. W. SAGE,“Recursive generalized-least-squ
dure for online identification of process parameters,” Proc. IEEE, Vol. 116 ( 1969), pp. 2057-2062.
K.L. SURYANARAYANAN, and S.G. RAO, “A unified [18] A.C. SOUDACK, to discrete-time systems identification,” Int’l. J. Control, Vol. 14, No. 6 ( 1971), pp. 1009-1029.
[ 191 J.M. MENDEL, Discrete techniques ofparameter estimation: The equation
mulation (New York: Marcel Dekker, Inc., 1973).
[20] S.J. MERHAV, and E. GABAY,“Convergence properties in linear paramet systems,” Identijication and System Parameter Estimation - Part 2, Proc Symp., ed. P. Eykhoff (New York: American Elsevier Publishing Co., In pp. 745-750.
Theory and practice of recursive ide [21] L. LJUNG,and T. SODERSTROM, (Cambridge, MA: M.I.T. Press, 1983).
[25] D.J. WILDE,Optimum seeking methods (Englewood Cliffs, NJ: Prentice H
[26] B. WIDROW,“Adaptive sampled-data systems,” in Proc. First Int’l. Fe Automatic Control, Moscow, July 1960, Part 1, pp. BPI-BP6.
[27] B. WIDROWand S.D. STEARNS, Adaptive signal processing (Englew Prentice Hall, 1983, chap. 12.
[28] S. HAYKIN, Introduction to adaptivefilters, (New York: Macmillan, 198
[29] S. HAYKIN,Aduprivefilter theory (Englewood Cliffs: Prentice Hall, 198
Theoryan [30] J.R. TREICHLER, C.R. JOHNSON, JR., and M.G. LARIMORE, adaptivefilters (New York: John-Wiley, 1987).
lters Pr [31] C.F.N. COWAN, and P.M. G ~ ~ ~ ~ , A d u p t i v e f i(EnglewoodCliffs: 1985).
[32] L.L. HOROWITZ, and K.D. SENNE,“Performance advantage of comple controlling narrow-band adaptive arrays,” IEEE Trans. on Circuits an Vol. CAS-28, No. 6 (June 1981), pp. 562-576.
[33] J.K. KIM and L.D. DAVISSON,“Adaptive linear estimation for sta dependent process,” IEEE Trans. Info. Theory, Vol. IT-21 (January 1975)
“Linear prediction: A tutorial review,” Proc. IEEE, Vo [34] J. MAKHOUL, 1975), pp. 561-580.
[35] L. J. GRIFFITHS, “Rapid measurement of digital instantaneous frequen Trans. Acoust., Speech, Signal Processing, Vol. ASSP-23 (April 1975), pp
[36] J.P. BURG,“Maximum entropy spectral analysis,” presented at the 37th An ing, SOC.Exploration Geophysicists, Oklahoma City, OK, 1967.
[37] L.D. DAVISSON, “Steady-state error in adaptive mean-square minimizat Trans. Info. Theory, Vol. IT-16 (July 1970), pp. 382-385.
“A rapidly converging first-order [38] T.J. SCHONFELD, and M. SCHWARTZ, gorithm for an adaptive equalizer,” IEEE Trans. Info. Theory, Vol. IT-17 ( pp. 43 1-439.
systems, 2nd ed. (Reading, MA: Addison-Wesley, 1990), chap. 8, Section
[42] T. KAILATH,Lectures on Wiener and Kalmanjltering (New York: Sprin 1981).
[43] B. WIDROW,J.M. MCCOOL,J.R. GLOVER,JR., J. KAUNITZ,C. W “Ada R.H. HEARN,J.R. ZEIDLER,E. DONG,JR., and R.C. GOODLIN, cancelling: principles and applications,” Proc. IEEE, Vol. 63, No. 12 I975), pp. 1692- 17 16.
4.0
INTRODUCTION
Adaptive plant modeling or plant identification is an important function for all adapti trol systems. In the practical world, the plant to be controlled may be unknown and p time variable. In order to apply adaptive inverse control, the plant, if unstable, must stabilized with feedback. This may not be easy to do, particularly when the plant d ics are poorly known. Empirical methods may be needed to accomplish this objectiv present purposes, we assume that the plant is continuous, stable or stabilized, linear a invariant. A discrete-time adaptive modeling system samples the plant input and out automatically adjusts its internal parameters to produce a sampled output which is match to the samples of the plant output when the samples of the plant input are used input to the adaptive model. When the plant and its model produce similar output s the adaptive impulse response is a good representation of the plant impulse response ality, the discrete-time adaptive model is a model of the samples of the impulse resp the plant, whose z-transform is designated as P ( z ) . The basic idea is illustrated in F where all signals and systems are considered to be sampled.
Ut
*
Adaptive model
Figure 4.1
jk
Fk(2)
Adaptive modeling of a noisy plant
output is Z k , given by zk
= yk
+ nk.
The discrete time index is k . The transfer function of the plant is P ( z ) . Its impulse response in vector f P = [ p , p 2 ' . 'P I ' ' . I T .
The components of this vector have values corresponding to the values of the impulses of the plant impulse response. The plant input is uk. The dynamic outpu is y k , the convolution of its input with its impulse response: yk = uk
* pk.
Taking z-transforms, this relation becomes Y(z) = C/(z)P(z).
The parameters of the adaptive model in Fig. 4.I are generally adjusted by an ada rithm to cause the error 6k to be minimized in the mean square sense. The desire for the adaptive model is Zk. A common and very useful form of adaptive model or adaptive filter is delay-line or transversal filter whose tap weighting coefficients are controlled b tive algorithm. This t y p of adaptive filter is well-kEown in the literature [2]-[4 Fig. 4.I , it converges to develop a transfer function P ( z ) which is an estimate of i_mpulse response vector of P ( 2 ) is represented by (4.2). The impulse response P ( z ) is represented by (4.5): A
pk = l j l k b 2 k
' ' '
There are n weights and each weight is a function of k , adapting iteratively. Othe adaptive filter can also be used, and they are well described in the literature 161-[9 ious adaptive filters become linear filters when their weights converge or otherwi fixed. Adaptive filters converge to approximate Wiener solutions when they are minimize mean square error. Wiener filter theory is useful in predicting asymp verged behavior of adaptive filters. Adaptive models can be generated that are very close repres5ntations of plants. At any time k , however, there will be differences between Pk and P . ferences will be called mismatch. There are three sources of mismatch.
1. One source of mismatch comes from representing a plant, whose impuls is really of infinite length, in terms of a model whose impulse response length.
weights. Only if the adaptive process were done infinitely slowly, using amount of real-time da,: would there be no weight noise. Fast adaptatio noise in the weights of P k .
Neglecting all of these limitations for the moment (they will be addressed proceed to examine idealized plant modeling in the presence of plant noise.
4.1
IDEALIZED MODELING PERFORMANCE
The following analysis of adaptive modeling is idealized in the sense that the adap is assumed to be of infinite length like the plant itself, and it is assumed that all p are excited by the statistically stationary plant input signal. This signal is ass persistently exciting [ 171 [20]. The adaptation process is assumed to have conv no noise present in the adaptive weights. The adaptive model has an input uk and it delivers an output j k , an esti plant dynamic output rekponse yk. When converged, the optimized transfer fun adaptive modekwill be P * ( z ) . Its impulse response in vector form will be design A formula for P * ( z ) can be obtained from Wiener theory as follows. Since the impulse response corresponding to P * ( z ) is by assumption not r be either causal or finite in length, the unconstrained Wiener solution, from
w.
where aUz(z) is the z-transform of the crosscorrelation function &(k),
k=-W
and where
(z) is the z-transform of the autocomelation function &,(k),as f
Assume that the plant disturbance nk is uncorrelated with the plant input u the plant dynamic output response yk. As such,
4uz(k)
= E[uj(yj+k
+ nj+k)l= 4uy(k).
P(z).The relation is @uy(z>
Substituting into (4.1 1) gives
= @uu(z)P(z).
P(Z)= P(z>.
We conclude that in spite of the presence of plant disturbance, the least squar model will, under the above assumptions, develop a transfer function equal to plant. This result will hold as long as the plant disturbance is uncorrelated wit input signal. The true desired response of the adaptive model is yk, but this is, of cours able. In Fig. 4.1, the training signal available to the adaptive process as a desire input is the disturbed plant output Zk. Training with Zk gives the same Wiener s would have been obtained if it were possible to train with Yk itself.
MISMATCH DUE T O USE OF FIR MODELS
In practice, the linear plant P(z)generally exhibits exponential transient behavio fore has an infinite impulse response (IIR). On the other hand the adaptive mod generally used here has a finite impulse response (FIR).' Refer to Fig. 4.2. Il the process of_modeling an IIR plant with an FIR model. The result is mismatc ence between P*and P.Wiener theory can be used to study this mismatch. Th is shown as an infinite tapped delay-line. Refer to Fig. 4.2. The Wiener solution (3.10) for the adaptive model can in this case as p* = [ 9uul-" 4uzJ. In more detail, formula (4.14) becomes 4 u u (0)
4uu(l)
... 4 u u ( n - 1)
4UZ
(0)
4 u u (0)
4uu(n - 1)
4uu@
-2)
...
4 u u (0)
'Adaptive plant modeling need not be restricted to an FIR approach. IIR modeling is certainly po both advantages and disadvantages. Adapting both poles and zeros is sometimes difficult, particular disturbance is present. An excellent discussion of IIR modeling is presented by John Shynk in [ 101
Plant
Figure 4.2
An
P
FIR tapped delay-line model for an IIR plant.
Assuming that the plant disturbance nk is uncorrelated with the plant input uk, E be used so that the last column of (4.15) can be replaced by a column of &,,'s. A
P" = [ &"I-"
A
+uvl.
This result is precise but generally hard to interpret on an intuitive basis. One special case, where the plant input signal ut is white, is of interest. H
Making use of (4.14) and using the fact that a white input has an autocorrelati in the form of a delta function, we may write
is white. When the plant input is colored but persistently exciting, the match will work similarly, but exactly how well can only be determined by making use theory. To obtain good dynamic modeling, it is necessary that the finite dura model’s impulse response span the plant’s impulse response up to the point whe decays to a “negligible” level. How long to make the model is always an uneas In a practical situation, one would need to have a rough idea of plant impulse re ration and input statistics in order to choose the number of adaptive weights n t mismatch error essentially to zero. The parameter n can always be made large enough to bring this error to z ever, from (3.71) we know that misadjustment is proportional to n . By making n weight vector becomes noisy as a result of adaptation and the only way of reducin is to adapt slowly. Slow convergence of the entire control system would be a resu takes care in the choice of n. An extensive literature exists concerning the determ the order of an unknown plant [ 1 1 1-[ 161.
M I S M A T C H DUE TO INADEQUACIES IN T H E INPUT SIGNAL STATISTICS: USE OF DITHER SIGNALS
To achieve a close match between the adaptive model and the unknown plant ov fied range of frequencies, the plant input uk needs to have spectral energy over th frequencies. If the plant input has uniform spectral density over the frequencies then error tolermre will be uniformly tightly held over this frequency range. In m however, the plant input uk fails to have adequate spectral density at all frequen good fit is required. The result is mismatch, the development of a difference b and P. Another form of difficulty that often arises in adaptive control systems re the plant input uk being nonstationary. Adaptive modeling processes do best with inputs. As an example, suppose that the command input were zero over a long time, then stepped up to a constant DC level and held for a long time, then dropped constant Dc level and held fora long time, and so forth. Level switching might be infrequent and sporadic, but when it takes place, one requires precise respons adaptive control system. Circumstances of this type present real problems for the process when the plant input is not dynamic and rich in spectral content but is by constant. One way to circumvent these difficulties lies in the use of random dither si to the plant input. The dither offers the advantage of providing a known input s
added to the controller output to form the plant input Uk. This is an effective sc the controller output is a stationary stochastic process and the independentdithe achieve a desired spectral character for Uk. Since the dither is an independentran its spectrum adds to that of the controller output to make up the spectrum of U an analogy here to the method of flash photography in full daylight, using the fl shadows. In contrast to this, schemes B and C of Fig. 4.3 use dither exclusively in e adaptive plant modeling process. The purpose is to assure known stationary s the input modeling signal. When the controller output is nonstationary, one m off not including it at all in the modeling process. Using scheme B with a white dither, the mean square error will be minim the impulse response of the adaptive model exactly matches that of the plant ov tion span of the model’s impulse response. The minimum mean square error w B is greater than when using either scheme A or the basic ditherless modeling Figs. 4.1 and 4.2. With scheme A, the minimum mean square error will be equal output disturbance power. With scheme B, the minimum mean square error will the power of the plant disturbance plus the power of the controller output u; afte ing through the plant. The higher value of minimum mean square error will un cause increased noise in the adaptive model’s weights. This is in accord with th misadjustment presented in Chapter 3. Otherwise scheme B is an excellent one It is possible to improve on scheme B, and with the cost of a somewhat inc tem complexity, one can reduce the weight noise while adapting with the same vergence and with the same dither amplitude as with scheme B. Thus, scheme C good features of scheme B and overcomes the drawback of having an increase mean square error. Plant modeling with scheme C is block diagrammed in Fig. 4A3(c). The b filter is the actual adaptive modeling filter. Its transfer function’ is A(z) and its sponse vector is&. Its input is the dither &. Scheme C is similar to scheme B an the same converged Wiener solution. The only difference is that scheme C inc technique for eliminating the effects of u; propagating through the plant P ( z ) in signal Ck. A filter is used which is an exact digital copy of pk(Z). Its input con controller output u; and its output is subtracEd from the plant output to provid response training signal for the adaptation of E k ( Z ) which contains no dynamic c originating with u;. Inclusion of the copied Pk ( z ) filter does not affect th_eWien Since its output is uncorrelated with the dither 6 k . Inclusion of the copied & ( Z ) fi
c(z)
*We define henceforth as the z-transform of the instantaneous values of the weight vector f tive model at time k.
nk
I
Plant
I
Controller output
Desired response
o
Plant disturbance nr
4
+
Controller output
Desired response Error
Dither
A d a p e e model
6k
pk(z)
P ou
Fk
Dither Scheme
(b) Figure 4.3
Dithering schemes for plant identification: (a) Dither scheme A: (b) Dither sche
4
Zt
Pla outp
Controller output
/
PPY
*
pk(Z)
Dither 6k
Dithe Schem
Controller output
o
input
(Alt Dither St
f (d)
Figure 4.3
Adaptive model
(continued). (c) Dither scheme C: (d) Dither scheme C (Alternative represent
ADAPTIVE MODELING SIMULATIONS
To demonstrate the process of adaptive modeling of an unknown plant, simulat ments have been performed and the results will be described here. For the purp perimentation, the plant will be known to us, but “unknown” to the adaptive alg Chosen for these modeling experiments is the unstable nonminimum-phase previously in Chapter 1, given by
- 0.5) + I)(s - 1) (S
(s
Since we know the plant, we can easily stabilize it. A feedback stabilize in Fig. 4.4. A root-locus plot is shown in Fig. 4.5. Two values of loop gain k we k = 2 1 and alternatively,k = 24. Both gains yield stabilized conditions. If we re transfer function of the stabilized system by G(s), the transfer functions for th values are for k = 2 1, 2 1(S - 0.5) G ( s )= (S - 3.7152)(s - 0.1324 0.9601j ) ( s - 0.1424 - 0.9601j ) ’
+
I I
Compensator
Plant
I I
I I I I
and fork = 24,
24(~ - 0.5) (s 1)*(S 2)’ The poles of these transfer functions are marked on the root-locus plot of Fig. 4. Choosing a compensator and loop gain to achieve stabilization was greatly by having knowledge of the plant poles and zeros. Without this knowledge, a gr experimentation would have been required for successful plant stabilization. G ( s )=
+
+
Figure 4.5
Root-locus plot for plant stabilization loop.
In an adaptive inverse control system, the controller will be discrete. A analog converter will be needed to take the discrete control signal and use it plant. For this experiment we choose the DAC to be a zero-order hold. We wi stabilized plant by including the DAC. The model of the stabilized plant will be we will need to sample the plant output for the adaptive modeling process. Th is illustrated by the block diagram of Fig. 4.6. The adaptive model, upon conve closely approximate the discretized G ( s ) ,that we will represent by G(z), defin
The operator 2 means: take the z-transform. The discretized transfer functions gain values are for k = 2 1, 0.906(~- 1.051 3 ) ( ~ 0.8609) '(') = (z - 0.9813 O.O945j)(z - 0.9813 - O.O945j)(z - 0.6897)' and fork = 24, 0.1032(~- 1.0513)(~ 0.8608) * G(z) = (Z - 0.9048)*(~- 0.8187) The impulse response corresponding to G(z) for k = 21 is plotted in Fi impulse response of the adaptive model, obtained in accord with the scheme is shown in Fig. 4.8. The input to the DAC was a random, zero-mean, first-or process. Learning is demonstrated by the plot of squared error over time, shown The error became very small, and the resulting adaptive model closely fits the ide of Fig. 4.7. A strong disturbance was added to the plant output (random, zero-mean order Markov) to test the ability of the modeling process of Fig. 4.6. Allowing e for convergence, the adaptive process was momentarily stopped, and a snapsho pulse response of the adaptive model was taken and this is plotted in Fig. 4.10
+
+
+
I I
Figure 4.6
s+ 1 (s+7)(s-2)
Compensator
*
Adaptive modeling of the stabilized plant.
I
1
Figure 4.7
Impulse response of discretized stabilized nonminimum-phase plant with k =
Figure 4.8
Adaptive model of discretized stabilized nonminimum-phase plant with k =
Figure 4.9 k = 21.
Square of the error versus iteration cycle number for the adaptive modeling process o
tortion in the impulse response is evident, but the similarity to the ideal impul is clear. The distortion disappears when p of the LMS learning algorithm is ma Reducing p 10 times reduces the distortion to an almost invisible level.
Figure 4.10 with k = 21.
Adaptive model of discretized stabilized nonminimum-phase plant subject to random
All of these experiments were run again for a different case, this time w The discretized transfer function G(z) is given by (4.25). The impulse respons in Fig. 4.14. Modeling the stabilized plant in accord with the scheme of Fig. 4.6, t impulse response that results is plotted in Fig. 4.15. The learning process is il the plot of Fig. 4.16 which shows the square of the error versus the number of cycles. Disturbance was added to the plant output (zero-mean, first-order Mark modeling process (with k = 24) was repeated. Allowing time for convergence of the adaptive model was taken and plotted in Fig. 4.17. Distortion in the adapt response due to plant disturbance is visible, but this goes away when p is made Using a dither signal to train the adaptive model in accord with scheme C shown in Fig. 4.18 was obtained (with k = 24). The block diagram of the trainin shown in Fig. 4.1 1. Comparing Fig. 4.18 with the ideal response of Fig. 4.14 de how well the learning process works. Adding a random, zero-mean, first-orderM turbance to the plant output, allowing enough time for scheme C to converge, a snapshot of the adaptive model’s impulse response, some distortion appears in t response, as shown in Fig. 4.19. This distortion diminishes as p is reduced in s appears when p is made very small. These experiments show that stabilized plants can be modeled with FIR a ters. When the natural signal driving the plant is adequate for modeling, the Fig. 4.6 can be used. When dither is necessary, either scheme A, B, or C cou Scheme C has been used in these experiments and it is diagrammed in Fig. 4.1 1. P bance does not bias the modeling solution but could cause the model’s weights To reduce the effects of the noise, one needs to adapt more slowly by reducin of F .
SUMMARY
In this chapter, several modeling methods were introduced which were based tion with available control signals, with dither signals, and with a combination o available signals. All of these methods h%ve their advantages and their limitatio all attempt to cause the adaptive model f k ( z ) to be a “good match” to the plan difference between the model and the plant is the mismatch which may be defi terms of transfer functions and weight vectors as Afk(z)
& &(z) - f (z);
A -
APk = Pk - P.
5+
2-
-
DAC
Figure 4.11
: I
s+ I (~+7)(~-2)
Compensator
/
(s+
I)(s-
I)
-Plant
Modeling the stabilized plant using dither scheme C.
I / I
II
-
Figure 4.12 Adaptive model of discretized stabilized nonminimum-phase plant with k = 21, dither scheme C.
Fipre4.13 Adaptive model of randomly disturbed discretized stabilized nonminimum-phase plan obtained with dither scheme C.
-0.1
-0.2 -0.3
Time (0.1 sec.) Figure 4.14
Impulse response of discretized stabilized nonminimum-phase plant with k =
-'-I'
Figure 4.15
Adaptive model of discretized stabilized nonminimum-phase plant with k =
Figure 4.16 k = 24.
Square of the error versus iteration cycle number for the adaptive modeling process
Figure 4.17 Adaptive model of discretized stabilized nonminimum-phase plant subject to rando with k = 24.
E"
d
-0 1 -0.2 -0 BL 3
20
40
60
80
120
140
Time (0. I sec.)
Figure 4.18 Adaptive model of discretized stabilized nonminimum-phase plant with k = 24, o dither scheme C.
20
40
60
80
140
Time (0.1 sec.)
Figure 4.19 Adaptive model of randomly disturbed. discretized. stabili7ed nonminimum-phase p 24. obtained with dither scheme C.
Weight noise is always present during adaptation, during initializingtrans steady state as the model weight vector randomly hovers about the Wiener solu when the adaptive process converges, the covariance of the modeling error is COV[AP~ =]
weight noise covariance matrix
I
‘
This covariance matrix is obtained for scheme C, with white dither, from Eq. ( Knowledge of the mean and the covariance of the mismatch will be use mining the mean and covariance of the inverse modeling error, which will subs related to the overall error of the adaptive inverse control system. This chapter introduces and explains dither schemes A, B, and C. They re tinctly different approaches to the use of dither signals for plant modeling. cal analyses of schemes A, B, C are presented in Appendix B. Further explana means of behavior is given there. A comparative analysis, citing advantages an tages of these schemes, is discussed. Performancecharacteristicsof these schem in Table B.3 of Appendix B. Scheme C is often the method of choice when us signal as an aid to the modeling process. For scheme C, it is shown in Appendix B that Stable range for p
Unique time constant
Weight noise covariance matrix weight noise
I.
Bibliography for Chapter 4 [ll
B. WIDROW,and M.E. HOFF, “Adaptive switching circuits,” in IRE Conv. Rec., R.4, pp. 96-104.
ed. R.E. Kalman and N. De Claris (New York: Holt, Rinehart and Wins pp. 563-587.
B. WIDROW, and E. WALACH, “On the statistical efficiency of the LMS with nonstationary inputs,” IEEE Trans. Info. Theory - Special Issue o Filtering, Vol. 30, No. 2, Pt. 1 (March 1984), pp. 211-221.
M. MORF,T. KAILATH,and L. LJUNG,“Fast algorithms for recursive tion,” in Proceedings 1976 Con8 on Decision and Control, Clearwater B December 1976, pp. 916-921.
L.J. GRIFFITHS, “A continuously adaptive filter implemented as a lattice in Proc. IEEE Int ’I. Con$ on Acoust., Speech, Signal Processing, Hartfor 1977, pp. 683-686.
“Recursive least squares D.T.L. LEE,M. MORF,and B. FRIEDLANDER, mation algorithms,” IEEE Trans. on Circuits and Systems, Vol. CAS-28, N 1981), pp. 467-481.
B. FRIEDLANDER, “Lattice filters for adaptive processing,” Proc. IEE NO. 8 (August 1982), pp. 829-867.
J.J. SHYNK,“Adaptive IIR filtering,” IEEE ASSP Magazine, Vol. 6. N 1989), pp. 4-2 1.
H. AKAIKE, “Maximum likelihood identification of Gaussian autoregressi average models,” Biometrika, Vol. 60 (1973), pp. 255-265.
H. AKAIKE, “A new look at the statistical model identificatio Trans. Auto. Control, Vol. AC-19 (1974), pp. 716-723. J. RISSANEN, “Modeling by shortest data description,” Automatica, Vol. pp. 465-47 1.
G. SCHWARTZ, “Estimating the dimension of a mode1,”Ann. Stars., Vol pp. 461-464.
M.B. PRIESTLEY, Spectral analysis and time series, Vols. 1 and 2 (New Y demic Press, 1981). S . HAYKIN, Adaptivejlter theory (Englewood Cliffs, NJ: Prentice Hall,
systems, 2nd ed. (Reading, MA: Addison-Wesley, 1990). Useful comm
subject of persistent excitation are given in chap. 8, Section 8.4.
5.0
INTRODUCTION
Control concepts taught in Chapter 1 involve the use of adaptive plant inverses as cont in feedforward control configurations. Pursuing these ideas, our next step is the de ment of general techniques for finding inverses of plants that need to be controlle shall restrict our development to apply only to stable plants. If the plant of interest stable, conventional feedback should be applied to stabilize it. Then the combination plant and its feedback stabilizer can be regarded as an equivalent stable plant. The s is discussed in detail in Appendix D. Only linear, single-input single-output(SISO) sy will be treated here. Nonlinear and MIMO systems will be discussed subsequentlyin ters 10and 11. The plant generally has poles and zeros. The inverse of the plant therefore shoul zeros and poles. If the plant is minimum-phase, that is, has all of its zeros inside th circle in the z-plane, then the inverse will be stable with all of its poles inside the unit If the plant is nonminimum-phase,then some of the poles of the inverse will be outsi unit circle and the inverse will be unstable. In general, not knowing whether the p minimum-phase or not, there is uncertainty about the feasibility of making a plant in This uncertainty can for the most part be overcome and excellent inverses can be m practice by using the appropriate adaptive inverse modeling techniques. We shall begin with a discussion of inverse modeling of minimum-phaseplant discuss inverse modeling of nonminimum-phaseplants. Model-referenceinverse mo will be described next. The effects of plant disturbance will then be considered, and and offline inverse modeling will be illustrated. Finally, the effects of gradient noise the plant inverse model will be analyzed, leading to expressions for noise in the w of the inverse model and for the variance of the dynamic system error of the entire c system.
5.1
INVERSES OF MINIMUM-PHASE PLANTS
Refer to Fig. 5.1. The plant is represented by P ( z ) . The plant inveLse, to be used ultim as the controller, shall be designated by C ( z ) if it is ideal, or by C ( z ) if it is obtaine practical way and is not quite perfect. Assume that plant P ( z ) is minimum-phase, h
! Plant inverse
Modeling signal
ClW
Error
I
I Figure 5.1
Forming a plant inverse..
Suppose for example that the plant has the transfer function
This plant is causal and it is stable since both of its poles are inside the unit z-plane. It is minimum-phase since its zero is also inside the unit circle. The r this plant is a perfect inverse,
The transfer function C ( z )can be expanded by long division, according to
3 -5 + . . . . + -z-2 3 - -3z - 3 + -2-4 3 - -z 2 2 4 8 16 It is clear from (5.2) that C(z) is stable since its pole is inside the unit circle. C ( a,s evidenced by expansion (5.3). Referring now to Fig. 5 . I , if the adaptive i C(z) had an infinitely long impulse response, it could perfectly reakze C(z). finite impulse response but a very long one, the difference between C ( z ) and be negligible. We will pause for a moment to de_monstrate that the unconstrained Wie for the adaptive filter transfer function C(z) will be equal to the inverse of P modeling signal in Fig. 5.1 be white, with unit power. The z-transform of this si correlation function is therefore unity. From Eq.(2.22). the t-transform of the autacorrelation function is P(z)P(z 1.
c(z)= 1 - -z-I 3
-'
Minimizing mean square error therefore provides the correct inverse.
INVERSES OF NONMINIMUM-PHASE PLANTS Next we consider the example of a plant whose transfer function is
This well-behaved plant is causal and stable, but nonminimum-phase because i outside the unit circle. The inverse of this plant is
It is evident that C(z) is unstable, since its pole lies outside the unit circle. Using s as an open-loop controller would be disastrous. But there is a way to alleviate th which is based on the theory of two-sided Laplace transforms. C ( z )can be expanded in two ways:
1-
+
iz-2
11 27 27 27 -z - -2 -2 8 16 32 64 128 1 +22-' The first expansion corresponds to a causal but unstable inverse. Not good. T expansion corresponds to a noncausal inverse, but at least it is stable. C(z) in eith forms would not result from Wiener optimization. The unstable form (5.9) w the mean square error to be infinite. Minimization of mean square error would this solution. The second expesion (5. lo), the stable one, is noncausal and th realizable by the causal filter C(z). The first two terms in Eq. (5.10) are causal and would be realizable. If the remainder of the terms were relatively small, then could be approximated by the first two terms and a causal filter would be able a useful approximate inverse. This is not the case with the present example ho the idea is suggestive. Before pursuing this idea, let us see what the causal Wien would be for this particular example. We will use the Shannon-Bode approach. illustrates how this is done. The modeling signal is assumed to be white, of zero C(z) =
3 --2
-I
--
+
+
+'..
z-lu +2z)
Modeling signal I
Optimized
Whitening filter
causal filter
L , - , - , , , - - - - - - - - - I
Causal plant inverse Error
Desired response Figurr 5.2
An example o f Shannon-Bode design of a causal Wiener plant inverse.
For the present example, using a white modeling signal. the z-transform correlation of the plant output is
After multiplying and factoring, the transfer function of the proper whitening tained as
Without regard to causality, the transfer function of the optimized filter tained by inspection of Fig. 5.2 as the inverse of the product of transfer funct plant and the whitening filter. This is
+
z-'(l 2z) ( 1 +2z-1) .
The next step is to find a stable impulse response corresponding to (5.13) and the noncausal portion. Making an expansion of (5.13,
i.
The minimum mean square error can be shown to be This is a very large m error compared to unity, the mean square value of the modeling signal. The cau filter will not make a very good controller. The difficulty comes from attempting nonminimum-phaseplant to respond instantaneouslyto the command input. In delayed responses can generally be accomplished more precisely, with much lo Plants having transport delay and other characteristics that fall under the g scription of nonminimum-phasecannot be made to respond instantly to sudden the command input. The best thing to do is to adapt the controller to provide de respo%es to the modeling signal. The idea is illustrated in Fig. 5.3. The delaye verse CA( z ) is adaptively chosen so that when its transfer function is multiplied b , transfer function o transfer function P ( z ) , the product will be a best fit to z - ~the of time delay.
Modeling signal
Delayed + plant inverse
,
L ( 2 )
We will rework the above example. Refer to Fig. 5.2 but incorporate a de the desired response signal path, as is done in Fig. 5.3. The plant stays the sam input and output stay the same, and so the whitening filter specified by (5.12) r same. Ignoring the requirement of causality for the moment, the transfer func optimized filter is obtained as the inverse of the product of the plant and white The result is transfer functions multiplied by
The minimum mean square error for A = 4 can be shown to be approxim This represents very low error. The new causal Wiener filter would thus make a controller. If a much larger value of A were chosen however, many more term ries would be included and an even more perfect delayed inverse would be the the delay in the overall control system response would of course be greater if filter were used as a controller. With infinite delay, the delayed inverse would be useless from a practical point of view. It is clear that increasing A reduces the mean square error. For any nonminimum-phaseplant, this would generally be th any minimum-phaseplant, A = 0 would suffice, except when the plant has more zeros, then A = 1 would suffice.' The above analysis is based on the assumptionLhat the adaptive inverse f is causal and has infinite impulse response (IIR). If C,(Z) is causal and of fin response (FIR), then increasing A beyond the point of any reasonable need cou ful,One would in effect cause the impulse response to be "pushed out of the tim of CA(Z).Figure 5.4 shows the impulse response for the plant represented by Figure 5.5(a)-(e) shows delayel inverse impulse responses for various values o plots have been developed for CA( z ) being an FIR tapped delay line with 11 we a broad middle range of A, from about 5 through 9, the optimized impulse re approximately the same shape except for delay. Outside that range, the choice o the shape of the impulse response. Figure 5.6 shows a plot of minimum mean s versus A for the above example. Experience with the formation of inverses for a variety of plants of wid complexity has shown that the shape of the curve of Fig. 5.6 is typical, with a w flat minimum. Choice of A is generally not critical within the flat minimum. W information, a good rule of thumb would make A about equal to half the chose ? A ( z ) . From there on one could experiment with A, generally keeping this pa small as possible, consistent with low adaptive inverse modeling error. The smallest value of A that minimized the minimum MSE plotted in Fig. A = 6, gave an excellent delayed inverse with extremely low error. Convolving impulse response with the plant impulse response provided the impulse respo cascade, and this is plotted in Fig. 5.7(a). If the inverse were perfect, this plot w zero response at all times except for a unit response at the chosen value of A, tha Imperfections in the inverse would be evidenced by small side lobes existing abo
I When the analog plant is discretized, more poles than zeros causes the discrete impulse response a delay of one sample period.
5
30 Sample time (k)
20
10
Figure 5.4
35
40
45
Plant impulse response.
spike at k = 6. Such imperfections are almost nonexistent in this plot. The step the cascaded plant and delayed inverse is shown in Fig. 5.7(b). The result is an fect step, delayed by 6 units of time. This is the step response of the correspond inverse control system. From the simple analytical example studied above, one can see the pos finding excellent delayed inverses for both minimum-phaseand nonminimum-p F e have used Shannon-Bode theory to determine optimal causal IIR transfer f CA( z ) and to determine associated values of mean square error. This theory can o when there exists a precise representation of the plant P ( z ) in pole-zero form world, P ( z ) would generally not be known. Instead of using Shannon-kode theo adaptive least squares algorithm would be used to determine the best C A ( Z )T. solution would closely approximatethe true Shannon-Boderesult. Although Sh works with IIR filters, the adaptive inverse filter would generally be an FIR filt filter architecture allows adaptation with the LMS algorithm, and there being no would be no stability problem.
MODEL-REFERENCE INVERSES
An adaptive process is illustrated in Fig. 5.8 for finding %model-reference p ? M ( z ) . The goal of this process is to obtain a controller C M ( Zthat, ) when u the plant, would result in a control system whose overall transfer function wo match the transfer function M ( z ) of a given reference model. The delayed inv ing scheme of Fig. 5.3 is a special case, where the model response is a simple d transfer function of z - ~ . To demonstrate how the model-reference inverse works, the following was done. Figure 5.9 shows the step response of an exemplary plant P ( z ) . It has transport delay, and it has a damped oscillatory response. Figure 5.10 sho
5
6
7
8
9
1
0
Sample time (k) (a)
Delay A = 3 0.5 -
L!
B -5 -M
I1
0
I
.-
0
-0.5-
5
6
7
8
9
1
0
Sample time (k)
(b)
I
I
5
6
7
8
9
1
0
Sample time (k) (C)
Figure 5.5 Inverse impulse response for various values of delay A. Inverse filter is FIR with m (a) A = 0 samples; (b) A = 3 samples; (c) A = 6 samples;
Delay A = 9
*
0.5 -
41
1
.
1
1
-0.5 -
1
Delay A = 12 0.5 -
4)
-0.5
-
-
-
.
1
1
,I
16
Modeling delay
Figure 5.6
Minimum mean square error versus inverse modeling delay A
response of a reference model M(z), designed to have the same transport dela an overdamped responseAFigure 5.10 also shows the step response of the cas and ) the plant P ( z ) . It can be seen that the casca model-reference inverse C M ( ( Z sponse matches very closely to the model step response, thus illustrating the ef of the model-referenceinverse process. For this experiment, the plant transfer fu
and the transfer function of the reference model was
For the inverse to have been accurate, it was necessary that the reference a delay at least as great as that of the plant, or to have a slow step-response rise ficient equivalent delay. As this condition was met, the step response of the co plant closely matched that of the reference model.
INVERSES OF PLANTS WITH DISTURBANCES
The inverse modeling process becomes somewhat more complicated when the p ject to disturbance. The inverse modeling scheme shown in Fig. 5.11, very mu of Fig. 5.8, is the simplest approach. Unfortunately,this approach does not wor with a disturbed plant. The reFon is that the plant disturbance, going directly in of the adaptive inverse filter C(z), biases the converged Wiener solution and p formation of the proper inverse.
3 &
e
-'2 E
w
0.80.60.4-
0.2-
I
Delay A = 6
I
0.8
w
&e
&
0.6 0.4
v)
0.2
-0.2'
5
10
20 Sample time ( k )
30
35
I
40
(b)
Figure 5.7
Response of cascade of plant and delayed inverse: (a) impulse response; (b) step
Figure 5.8
Finding model-reference plant inverse.
Without plant disturbance, the Wiener solution would be
This would be the result that we seek. Equation (5.22)is in accord with Eqs. (2.30 generalized for the model-reference case. Note that yk is the plant output wit bance (as in Fig. 5 . 8 ) and dk is the reference model output. The situation is different in the presence of plant disturbance. In Fig. 5. turbed plant output is .?I.. The Wiener solution can be written as
Since the plant disturbance nk is uncorrelated with dk and z k , Eq. (5.23) can be
The second term in the denominator causes a bias so that, for the disturbed plan
The inverse modeling scheme shown in Fig. 5.12 overcomes the bias proble ing use of an adaptive model of the plant. Instead of findin& a model-referenc the plant P ( z ) , the model-reference inverse is taken from P ( z ) . The idea is th essentially the same dynamic response as P ( z ) but is free of disturbance. A clo imation to the desired model-reference inverse, unbiased by plant disturbance obtained. It should be recalled from a discussion in Chapter 4 that plant d$urbance d fect the Wiener solution when the plant is directly modeled to obtain P ( z ) . Plant d does affect the Wiener solution, however, when doing inverse modeling by the Fig. 5.1 1. The inverse modeling scheme of Fig. 5.12 has been tried and tested wit plants and found to work very well. It should be Eoted, however, that the existe levels of plant disturbance forces one to adapt P ( z ) slowly to keep noise in t
6
8
10
12
14
16
18
20
Sample time (k) Figure 5.9
Step response of the plant used in the example of model-reference contro - - -
-
6
8
10
12
14
-
16
18
20
Sample time (k)
Figure 5.10
Step response of the controlled plant (e) superimposed on the desired step response of model, illustrating successful adaptation.
I Figure 5.11
Reference model
M(Z)
dk
An incorrect method for inverse modeling of a plant with disturbance.
Plant disturbance nk
uk
b
Model-reference inverse
6Z)
I
Error
Reference model M(z)
dk
fk
Plant disturbance nk
I
I I I I I I I I I I
I
+*
I
I
I I I
Modeling signal
I I
M(z)
I
dk
I I I I I I I
c-,-,-,,-,,-,,,-,,,,,,,I
Offline inverse modeling Figure 5.13
An offline process for inverse modeling of a plant with disturbance.
This scheme uses one of the methods of Chapter 4s obtain P^(z). An e copy of P^(z) is then-used in an offline processLo obtain C ( z ) . Computer-gen drives the copy of P ( z ) , and its output drives C ( z ) . The y e noise drives th model M(z),and its output is compared with the output of C ( z ) to obtain an e E ( z ) is adapted to minimize the mean square of this error. The offline inverse modeling process_canbe run much faster than real tim principle, that as instantaneous values of P ( z ) are obtained from the direct ms cess, the offline inverse modeling process provides corresponding values of C tially instantaneously.
EFFECTS OF MODELING SIGNAL CHARACTERISTICS THE INVERSE SOLUTION
Assume that ? ( z ) has an impulse response that is not constrajned, that it is allow an infinite impulse response in both directions in_time. Then C ( z )can be adjusted to C ( z ) ,assuming the availability of a perfect P ( z ) = P(z1. As such, any mod that has at least some energy at all frequencies will cause C ( z ) to be equal to C other hand, when ?(z) is restricted to be causal and FIR,not enough weights, o freedom in ? ( z ) are available to perfectly match C ( z ) .Under these conditions, shale of the modeling signal could have a considerable influence on the frequen of C(Z). In general, the simplest choice of modeling signal would be white noise of this choice can be visualized in the following way. Refer to Fig. 5.13. There will in general be a diffzence between the fre sponse curve of M ( z ) and that of p ( ~?)( z. ) . Optimizing C ( z )with a white mod will cause transfer function differences to be weighted equally at all frequencies area of the difference of the two frequency response curves to be minimized. U white modeling signal causes frequency response differences to be weighted m at frequencies where the modeling signal has higher power density. The modeling signal characteristics should be chosen to be like those of th input signal of the entire control system. The controller should be trained to prod overall control response. If no a priori knowledge of the command input signal i one could simply assume a default position by choosing a white, zero-mean mode
INVERSE MODELING ERROR
Thus far we have discussed various forms of inverse, the plant inverse itself, plant inverse, and the model-reference plant inverse all for unknown plants wit out plant disturbance. In the disturbed plant case, we have assumed that an ex% the plant P ( z ) is available to the inversion process. In reality, the plant model & be available instead. The question is, what effect does mismatch in & ( z ) , caused by noise in t weights, have upon ?k(z)? The mismatch APk(z) is related by
F ~ ( z =) P ( Z )+ A P ~ ( Z ) , h
Pk = P + APk.
or
IL
covariance matrix
J
where B is defined as the individual weight noise variance. The weight noise matrix is a scalar matrix. The weight noise components have equal power and a lated from weight to weight. Various modeling schemes were suggested to obtain F(z), with and with Without dither, the weight noise covariance matrix is obtained from Eq.(3.60). W using schemes A, B, or C, expressions for the weight noise covariance matrices derived and can be obtained from Appendix B. These expressions are all close mated by (5.28) forzmall p. The errors in P(z) at each time k cau? errors in ?k(z) at each time k . The verse modeling process of Fig. 5.13 causes Ck (z) to instantaneously track and fo Our present objective is to find the relationship between these errors. The erro have been characterized by
G(z)= C(Z)+ AWZ),
or
When ?k(z) is obtained in actual practice by inversion of &Q),ACk contains effects which give it a bias, plus noise resulting from noise in Pk. Assume that t are small and additive. The covariance of ACk will be = COV[AC~]
1
a function of COV[AP~]
The mean of ACk will be assumed to be E[ACk] = (truncation effects) = 0. A
The reason for Eq. (5.32) isLhat, in practice, M(z) includes enough smoothing and the impulse response of Ck(z)is chosen to be long enough to keep truncation small. It is useful to relate ACk(z) to APk(z). Refer to Fig. 5.13 and to its offl modeling process. Recall that the ideal inverse would be
Since ?k(z) will be used as the controller of the plant P ( z ) in a feedfoward ca figuration (illustrated in Fig. 1.3). it is possible to use the above expression for determine the overall error of the plant control process. This determination is o jective.
CONTROL SYSTEM ERROR DUE TO INVERSE MODE ERROR
Successful control will give the cascade of ?k ( z ) and P ( z ) a dynamic response matches the dynamic response of the reference model M ( z ) .Figure 5.14 shows t and M ( z ) , both driven by the command input l ( z ) of the control system. The in the outputs of P ( z ) and M ( z ) may be called the dynamic system error ~ ( z dynamic error in the sense that, all else being fixed, it is proportional to the com I ( z ) . If ACk(z) were zero, the dynamic system error would also be zero. In g dynamic system error is
[
=~
( z=) - I ( z ) . AC,(z) . P ( z ) .
Command Input I(<)
Reference rntdel
+-
M(2) Figure 5.14
Diagram for the definition of dynamic system error.
D sys
This is the transform of the dynamic system error. It can be thought of as the ou slowly time variable system which is pictured in Fig. 5.15. The command input plied at point A in this figure, and a filtered version appears at point B. This in tur to the filter APk(z) whose weights slowly fluctuate over time k, uncorrelatedly other, each weight having zero-mean and variance B. Of interest is the variance This is equal to the variance at point B multiplied by # multiplied l b i the number n of APk(z) (equal to the number of weights of the plant model P ( z ) ) . The tr the autocorrelation function at point B is
Command input I(z)
A
M(z) P(Z)
B
+ A&(z) C
*
Dyna system
Random' weights Figure 5.15
Construction of dynamic system error due to fluctuation in P ( z ) .
The variance at point B is @BB(O). The variance at point C is therefore
The system designer can predict this error variance given some notion of the com I ( z ) and of the plant characteristics P(z)_:The reference model-response M ( z of the The adaptive weight noise variance B of P ( z ) is the product of L./ and ni,& eling process, in accord with Eqs. (A.39) and (B.54). In the above discussion, we have assumed that both (2and ?k(z) were c enough so that there would be no truncation effects. Making pk(z) short would geous, as long as there would be no truncation effects. This is clear from Eq.(5 the dynamic system error variance to be pr-portional to the number of Pk ( z ) .In this derivation, it has been assumed that Ck ( z ) is very long, essentiall
6
cows
yhere min(m, n) means to use the smaller of the number of weights, m of ?k Pk(z). This result has been checked by simulation studies and it works very w ample will be presented in the next section. Equation (5.40) is simple and easy to use. The variance j? is the product of and emin is equal to the variance of the plant output noise when modeling witho when modeling with dither schemes A-or C. Sckeme B would require a special By knowing the number of weights in P (z) and C ( z ) ,one can determinemin(m, ~ B B ( Ois) the variance resulting from the command input going through
In the actual control system, this is essentially equal to the variance resulting fro ing through ? ( z ) . This is in fact the variance of the control signal applied to th
A COMPUTER SIMULATION
in order to verify JQ. (5.40), we have performed a computer simulation of th Fig. 5.14. The plant was chosen to be
The reference model was chosen to be M ( z ) = 1.
The plant controller ?k (z) was computed as follows: a. The plant model C of Fig. 4.3(c).
g(z)was obtained by an adaptive process using dither
a.
b. The optimal controller &(z) was computed algebraically at each time k using (5.34).
For each time k, the weights of t?k(z) were used in order to compute th system error. Various simulation parameters were chosen as follows:
a. The command input was a zero-mean white signal of unit power, E [ i z ] =
b. The dither was a zero-mean white noise of unit power, E [ S 3 = 1.
c. The adaptation constant of the process for finding & ( z ) was set at p = 0
ones obtained from Eq. (5.40). In order to use (5.40),we needed values for /3 and for [ @ ~ ~ ( 0 The ) 1 . value product of @ and the minimum mean square error in modeling for P ( z ) . In our tmin
=E I ~ ~ I ,
which was a variable parameter, and /3 = /Atmin = (0.001)E[n:].
The value of [@BB(O)] was approximated as the output power of the idea C(z) fed with the command input. For this case, the ideal controller was
The impulse response of C(z) was
{
0, 1,
k
l Since the command input signal was white,
ck =
sum of squares of impulses of impulse response of C(z)
= 1.
Using (5.44)and (5.47),the variance of the dynamic system error was estimate choice of the design parameters. The results are summarized in Table 5.1. For a of choices of /3, m ,and n, the simulation results matched the theoretical prediction about 25 percent.
EXAMPLES OF OFFLINE INVERSE MODELING OF NONMINIMUM-PHASE PLANTS
The unstable nonminimum-phase plant described in Section 4.4was stabilized back. The block diagram of the stabilization loop is shown in Fig. 4.4.Two
lo-’ 10-5 10-5 10-4
10-4 10-4
50 10 50 10 50 10 50
10 50 50 10 10 50 50
0.000100 0.000100 0.000500 0.00100 0.00100 0.00100 0.00500
0.000063 O.ooOo74 0.000605 0.00094 0.00104 0.00076 0.00542
studied in Chapter 4, with k = 2 1 and k = 24. The discretized impulse respons bilized plant is damped and oscillatory for k = 2 1 (see Fig. 4.7). This impulse much less oscillatory for k = 24 (see Fig. 4.14). For both of these cases, inve ing was done using the offline method illustrated in Fig. 5.13. Results are prese section. For k = 21, the discretized transfer function G ( z ) of the stabilized plant Eq.(4.24). A 400-weight adaptive model of G ( z )was made, and its impulse shown in Fig. 4.8. A delayed inverse of this impulse response has been obtain delay of 50 sampling periods, a 100-weight inverse has been taken, and the re pulse response is shown in Fig. 5.16. To demonstrate that the inverse is really its impulse response in Fig. 5.16 was convolved with the true discretized impul of the stabilized plant, shown in Fig. 4.7. The result is very good. The first 100 the convolution are shown in Fig. 5.17. A result almost as good could have been obtained with substantially les 10 sampling periods rather than 50. The reason for this can be determined from of Fig. 5.16. The inverse impulse response is almost zero until about 40 sampli 10 sampling periods before 50. Noise was injected into the plant to create the effect of plant disturban weight adaptive model was made whose impulse response is shown in Fig. 4. tion in the model was caused by the disturbance. This distortion got smaller as adaptation was made smaller. Using the distorted model of Fig. 4.10, a 100-we with a delay of 50 sampling periods was taken. This inverse impulse response Fig. 5.18. Convolving it with the true discretized impulse response of the stab yields the result shown in Fig. 5.19. This is not a perfect delayed unit impulse. A inverse is far from perfect, it is still highly effective as a controller. This subject i in Chapter 6. Changing the gain in the plant stabilization loop to k = 24, the discretiz function G ( z ) of the stabilized plant is given by Eq. (4.25). A 150-weight adap of G ( z )was made, and its impulse response is shown in Fig. 4.15. With a delay pling periods, a 100-weightinverse of the 150-weight model of G ( z )has been
20 -
J
-I
-20 -30
Figure 5.16
20
10
30
60
40
70
90
80
Time (0.1 sec.)
Delayed inverse of discretized stabilized (k = 21) nonminimum-phase plant mode
0.8-
0.6
-
L
-0.2
Figure 5.17
1
0
2
0
3
0
4
0
8
0
7
0
8
0
9
0
Time (0.1 sec.)
Convolution of true G ( z )fork = 21 with delayed inverse of its 400-weight model
-10-
-20.
-30L
10
x)
30
40
8
0
7
0
8
0
9
0
Time (0. I sec.)
Figure 5.18 Delayed inverse of discretized stabilized ( k = 21) nonminimum-phase plant model turbance (Fig. 4.10).
08-
06-
s0
.-
04-
E
Q
02-
-02 '
10
20
30
40
8
0
7
0
8
0
9
0
Time (0.1 sec.)
Figure 5.19 Convolution o f true G ( z )for k = 2 I with delayed inverse of its W w e i g h t model af disturbance (Fig. 4.10).
20 -
10 -
-10
-
-20 -
-30'1
Figure 5.20
20
30
40
80
70
Time (0.1 sec.)
80
90
Delayed inverse of discretized stabilized (k = 24) nonminimum-phase plant model
-0.2L
Figure 5.21
10
10
20
30
40
60
Time (0.1 sec.)
70
80
90
Convolution of true G ( z ) for k = 24 with delayed inverse of its 150-weight model
SUMMARY
An adaptive least squares algorithm can be used to set the weights of a filter to a the inverse of an unknown plant. If the plant is minimum-phase, an accurate inv readily made. If the plant is nonminimum-phase,an accurate inverse can be mad a delayed response. The technique can be easily extended to form model-referen so that the plant and inverse when cascaded have a transfer function that close that of a selected reference model. The inverse filter can be used as a controller plant. Generally, physical plants have disturbance. Adapting to form a plant i model-reference plant inverse in the most straightforward way results in a bia when the plant is disturbed. Techniques to overcome this difficulty were prese chapter. The inverse was based on a model of the plant which was not biased b disturbance. Gradient noise in the weights of fi (2) causes noisein the weights of ( latter is obtained from the former. Noise in the weights of ck (z) causes error in t response of the overall control system. Analysis of this error has resulted in the
ck
variance of dynamic system error
= [ $ B B ( O ) l . min(m, n) . B.
The system designer may not have all of the factors of (5.40) exactly, but in would have some idea about the sizes of these elements and would therefore be a a good estimate of the variance of the dynamic system error. This is important i mination of overall system performance. Equation (5.40) was checked with rega ulated adaptive inverse control system, and experiment and theory agreed in a w of cases to within 25 percent. In the next chapter, we shall examine various ways of using the inverse pl
10
3._-2 a
-
-10
I , . .
-
-20 -
-30
Delayed inverse of discretized stabilized (k = 24) nonminimum-phase plant model turbance (Fig. 4.17).
Figure 5.22
0.8 -
0.6-
4
-2 .-
Y
d
0.4
-
0.2-
-0.2
J
10
20
30
40
60
Timep0.I sec.)
70
80
90
Figure 5.23 Convolution of true G ( z )for k = 24 with delayed inverse of its 150-weight model affe disturbance (Fig. 4.17).
6.0
INTRODUCTION
An adaptive inverse control system is diagrammed in Fig. 6.1. If the controller wer its transfer function would be
Plant disturbance nk
Plant input
az)
Con_troller +
ik
-
*
uk
Pla out
2
Reference model M(z) Figure 6.1
An adaptive inverse control system that works well but adapts slowly.
The adaptive controller will generally not be ideal; its transfer function can there designated as
?(z) = C(z)
+ AC(z).
The controller weight vector can be expressed accordingly as =C
+ AC.
tor which is obtained one component at a time, and each gradient component is o measuring MSE with the corresponding weight increased and held for some tim creased and held for some time. The LRS algorithm tries random changes in the w tor. After each trial change, the MSE is measured and compared with the measure fore the trial change. The actual weight vector change is made equal to the trial c tiplied by the MSE difference (before and after the trial change). If the trial cha an improvement in performance, a lowering of MSE, the actual weight change w trial direction and proportion to the improvement. If the trial change causes a r performance, then the actual change will be opposite in direction to the trial di proportional to the reduction in performance. The DSD and LRS algorithms pe ilarly, except that LRS converges twice as slowly as DSD when both adapt wit level of misadjustment. Both of these algorithms converge extremely slowly co LMS when they are all set to adapt with the same level of misadjustment. It would be desirable to use the LMS algorithm because it is much faste and DSD. It cannot be used directly because the available error 6 k of Fig. 6.1 referred to the plant output.' LMS really needs an error referred to the plant in to the adaptive controller output. To get an appropriate error for LMS impleme would need to apply 6k to the inverse of the plant P ( z ) , thus requiring the solut to get the solution. The system of Fig. 6.1 is not our system of choice. In order to be able to make use of the LMS algorithm and other high-spe processes, the inverse modeling configuration of Fig. 6.2 has been devised. Th its &verse model are commuted, so that the error 6 k is directly available for the of C(z). Once F(z) is obtained, an exact digital copy can be used as a contro plant. This adaptive control system concept was first proposed in reference [2]. The system of Fig. 6.2 works very well as long as there is no plant_dist plant disturbance is present, its effect is to bias the Wiener solution so that C(z) a proper controller. The disturbance that appears at the plant output adds a comp covariance 0; the input signal of the adaptive inverse model, directly affecting solution for C(z). So what should one do? There are a number of choices, and th indicated in Fig. 6.3 offers the possibility of rapid adaptation and proper control presence of plant disturbance. The control system of Fig. 6.3 is bascd on the inverse modeling scheme o It works in the following way. A model Pk(z) of the plant P ( z ) is formed,_u case dither scheze A. An offline process can be used to obtain controller Ck digital copy of Pk(z), and the reference model M ( z ) . The offline process, il I Nor can
any of the exact least squares lattice algorithms be used in this application for the same
Reference model M(Z)
Figure 6.2 An adaptive inverse control system with commuted plant and inverse model. It works w plant disturbance has low level.
Dither
Plant disturbance
ak
nr:
@
Controller
I I I I
‘-4-p
Reference model
5 - , , - - , , - , , , - - - - - - - - - - -
j I
OfRine p m e s c
Figure 6.3 An adaptive inverse control system with offline inverse modeling adapls rapidly and w with plant disturbance
result is a controller and plant having an overall dynamic response which close mates the optimal dynamic response of the reference model M ( z ) . The offline process of FigSA6.3forms a model-reference inverse of the p p k ( Z ) . We have used the model & ( z ) rather than the plant P_(z) because the ou real P ( z ) is generally corrupted by plant disturbanc5 Since Pk ( z ) does noiperfe P ( z ) at all times, use of P ( z ) indetermination of C ( z )causes errors in C ( z ) . if p(z)were perfect, there would generally be limitations preventing the perfect of C ( z )= M ( z ) / P ( z ) . These limitations will be explored n_ext. We will assu a perfect = P ( z ) and proceed to find best inverses of P ( z ) . Errors in th process will be analyzed. Then the additional effects of errors in 6.2)will be c
c(z)
6.1 ANALYSIS
Figure 6.4 is a block diagram of an adaptive inverse control system that we wi for purposes of analysis. The method of plant modeling used here is dither schem block diagrams could be drawn incorporating other modeling methods, with dither, but we have selected scheme C for presentation and analysis because we b this scheme is likely to be a very useful one in the future. An offline process shown in Fig. 6.4 for the calculation of controller & copy of the plant model Pk(z). As discussed above, this offline process could b an adaptive agorithm using a modeling signal (which could be white or colored equivalent11 Ck ( z )could be found by an analytic process such as Levinson’s alg A copy of ck(z) is shown serving as the plant’s controller. The reference model M ( z ) is used in Fig. 6.4 in the offline inverse modeli Another copy of this reference model could be used symbolically for system erro ing. The dotted portion of the system block diagram suggests this possibility. T system error is E ( z ) , and this consists of a sum of four components: i. Plant disturbance ii. Dither noise filtered through the plant
*When given pk(:) and M ( z ) ,the offline process finds Pk(Z). Using an artificial mtdeJng sign process, if it is an adaptive one. can work at much faster than real-time rates to determine Ck(Z). It time is so short that it does not affect the convergence time of the entire control system.
? I fthe modeling signal is white. E t ( z )= M ( z ) / p t ( : ) .e 4 ( z ) could be obtained algebraicrlly. w tion. However, when the modeling signal is not white or when P ( : ) is nonminimum-phaseor when realized as an FIR filter, a Wiener solution is needed. An adaptive solution is easy to get. An alge could also be easily obtained if a computer were available to calculate “R” and “ P ” and to implem
M(z) : ..............
I
Plant disturbance
I
nk
0
I
, Command input signal [(z)
9.i-
.4-.) .
j
I
Controller
Dither 81. (white noise)
White modeling I' signal
.-l--+-H-h c_.PY
I
I I I
'q-i-'
6Z)
I
c-,--,--,---,,,-,--,,,,
Offline process for finding
Figure 6.4
i
Reference model
e(z)from p(z)
A model-reference adaptive inverse control system.
- ,
-
Component (ii) is noise at the plant output due to dither which, if white given by dither noise at plant output
)
sum of squares of impulses
=
w:1. [ of impulse response of
f(z)
]
Component (iii) is the system error at the plant output due to truncation e assumedio be small or negligible as a result of !a appropriate choice of weights n of fk(z), choice of number o f weights m of CI,( z ) , and an appropria the reference model M ( z ) . Component (iv) is the dynamic system error at the plant output, and its pow by Eq. (5.40). The overall system error power is the sum of the above described compo it is
The first term in (6.6) is the plant disturbance power E [ n : J , which is inde the design parameters of the control system. Of special interest are the last tw (6.6), which together comprise the excess error power due to imperfection of system:
( ) A [e c
:
power
~
ElS:].
pi]
I
+ min(rn, n ) . B
[GAH(0)1.
=o
Minimization of this excess error power is the next objective. The control system of Fig. 6.4 has only a small number of adjustable param the sampling rate is fixed and the numbers of weights m and n are cho2en. Th adjustables are the value of p in the LMS algorithm used in obtaining fk(z) an of the dither signal E [ S i ] . The time constant of the adaptive modeling proces is actually the learning time constant for the entire system. This time constan chosen to be as long as possible, but not so long that the adaptive process would to keep up with the natural variations in plant characteristics. Some knowledge of change of pl2nt characteristics would be helpful in fixing the value of the ad constant T for f ( z ) .
and the derivative is set to zero. The best choice of dither power is4
E[$I,
=
J
E[n:l[4~~(O)l . m i n h , n) 27 Piz
cp",
The two terms of (6.7) are equal under these conditions:
The two sides of this equation are identified with the plant output power due to and the plant output power due to dynamic system error. Accordingly, with d optimized,
(
dither noise powerat plant output
)
1
=
system error ( dynamic at the plant output
4
.
This is the criterion for optimality for the dither power when using scheme C. Th value of (6.8) is obtained by using the optimal dither power. The result is
(
=
Figure 6.5 is a generic plot of overall system error power versus dither powe Eq. (6.8).
COMPUTER SIMULATION OF A N ADAPTIVE INVERS CONTROL SYSTEM
In order to demonstrate the workability of an adaptive inverse control system, do eling with scheme C, gain experience with the optimization of dither power, Eqs. (6.8)-(6.12), a computer simulation of the system of Fig. 6.4 was perfo
4An exception to this arises when the plant disturbance has low power. requiring low dither po with (6.9). The use of low dither power in turn forces a large value of p , in accord with (B.34). M could cause instability. (See(B.50)].The maximum p in such cases would be limited by stability c Having thus chosen p and r, the dither power would be determimed by (B.34) rather than (6.9). In system output error power is obtained from (6.6).
-9
-6
Figure 6.5
0 Oplimal
+3
+6
+9
Dither power (dB)
Excess error power versus dither power.
system worked very well and proved to be self-contained and quite useful. Th worked well also. The following is a description of the experiment performed. A minimumwas selected for this test,
The plant disturbance was white, with zero-mean and power of 10, E [ n i ] = 10 ence model was chosen to be M ( z ) = 1. The command input was a zero-mean w of unit power. Sckeme C wakused for modeling the plant, and instead of using process to obtain C ( z )from P ( z ) and M ( z ) , an algebraic method based on equa was used. The simulation was run with different values of adaptation constant different values of dither power E[S;]. The time constant of the adaptive process for c ( z ) depends upon the pa and E [ S ; ] . These parameters were chosen so that in all cases, the adaptive tim was 5,000 samples. Accordingly,
There were 10 weights for p(z),and 20 weights for p(z).Each simulat run for at least three time constants before excess error was measured. Theoretically, the optimal dither power was determined by (6.9). To hav equation, we needed to know E [ n z ] , [ # B B ( o ) ] , min(rn,n), r , and x z o p : . We of this, since E [ n i ] = 10, and min(rn, n) = 10 because rn = 20 and n = 10, and p' in or to have the value of 5,000. We needed only to find [ # B B ( o ) ] and used the formula.
czo
impulses of its impulse response had been calculated, and this was then had what was needed to apply (6.9). Accordingly,
Ezp'
=
From (6.14), we obtained, the optimal adaptation constant as 1 1 p=-= 2.5(10)-4. 25 E[S,2] 2(5000)(0.4) From (6.12), the minimum excess error power was
=
/(A)
(10)(4.12)(0.26)(10)
= 0.207
In order to verify (6.16), (6.17), and (6.18), the control system was operat timal dither power in accord with (6.16). Then dither power was increased by 2, and then by a factor of 4. Returning to optimal, the dither power was then re factor of 2 and further by a factor of 4. The excess error power was measured in The results are given in Table 6.1. EXCESS ERROR POWER AS A FUNCTION OF DITHER POWER
TABLE 6.1
Dither Power [dB] -6 -3 Ooptimal
3 6
Excess Error Power (from Eq. (6.7)) Theoretical [dB] Experimental [dB] 2.12 1.25 I I .25 2.12
1.9 1.44 0.92 1.34 2.13
Referring to Table 6.1, it may be noted that the dither power is given in d theoretically optimum dither power of 0.4 represented by 0 dB. The other dithe the experiment differ by 3-dB increments. Theoretical and experimental value
The stabilized nonminimum-phase plant that was described in the previous chapt adaptively modeled, and inverse controllers have been obtained for values of th tion loop gain set at k = 21 and k = 24. The plant modeling was done without d using the command input signal. In this section, we will use simulation techniqu several adaptive inverse control systems involving the same plant. We will stu ability and control effort. Figure 6.6 shows the convolution of the adaptive inverse impulse respon true discretized impulse response of G ( z ) ,the stabilized plant, with k = 24. Th no disturbance. The inverse filter had 100 weights and a delay of 50 sampling p though the plant was nonminimum-phase, the inverse filter was an excellent on idenced by the convolution plot, an almost perfect unit impulse delayed by 5 periods. A comparison of the plant output with the command input delayed by 5 periods is shown in Fig. 6.7. The command input was a zero-mean first-order M cess. Excellent tracking can be seen. The control effort, the input signal to th shown in Fig. 6.8.
-0.2
10
20
30
40
60
Time (0.1 sec.)
70
80
90
Figure 6.6 Convolution of impulse response of true G ( z ) for k = 24 with adaptive delayed inv response. The plant had no disturbance.
.-
Y
d
-0.2
-0.4
-0.6
2
0
4
0
6
0
8
0
120
140
180
180
Time (0.1 sec.)
Figure 6.7 Tracking performance: A comparison of the plant output with the command input sign 50 sampling periods.
20
40
60
80
120
140
160
Time (0.1 sec.) Figure 6.8
The control effort: The plant input signal.
180
for the plant without disturbance. To make this comparison, the plant disturbanc subtracted from the plant output. Control of the plant disturbance itself is the Chapter 8.
-0.2L
10
20
30
40
60
Time (0.1 sec.)
70
80
90
Figure 6.9 Convolution of impulse response of true G ( z ) for k = 24 with adaptive delayed inv response. The plant was subject to disturbance.
Good dynamic control of the plant is therefore feasible even when c^ is n Figure 6.1 1 shows a plot of the control effort. The amplitude of this effort was same as that for the case of the plant without disturbance. This is not s%surprisin dynamic-control of the plant is done about as well with the imperfect C as with perfect C. One reason for using a smooth reference model M ( z ) is limitation of the of the control effort. When M ( z ) is a simple delay, the system step response is achieve such a sharp response, a strong control effort may be required. This cou nonlinear overall response and could sometimes even cause physical damage. To explore the effects of smoothing in M ( z ) upon control effort, further s were done. M ( z )was made low-pass with a delay by cascading a one-pole filter w of 50 sampling periods. Figure 6.12 shows the impulse response of M ( z ) and th response of ? ( z ) convolved with that of G ( z ) . The fit is not perfect, but very c
x 20
40
60
80
120
140
160
180
Time (0. I sec.)
Figure 6.10 Tracking performance: A comparison of the plant output (with plant disturbance su the command input signal delayed by 50 sampling periods.
I
-4! -6
20
40
Figure 6.11
60
80
, 120
140
180
Time (0.I sec.)
The control effort: The plant input signal.
180
- 0 0 2 L - L 10
20
-.1-
30
----.I-
40
Time (0 I sec
6 )
I
0
7
0
1 1
8
0
9
0
Convolution of impulse response of G ( : ) for k = 24 with impulse response of md inverte C ( 2 ) The plant had no disturbance The dotted curve i s the impulse response of M ( z )
Figure 6.12
-0 5
1
20
J 40
' 60
80
A ' . I . -
120
140
160
180
Time (0.I sec.) Figure 6.13
Tracking performance: A comparison of the plant output with the reference mod
O,./ 0.6
20
40
Figure 6.14
60
80
120
140
160
180
Time (0.1 sec.) The control effort: The plant input signal.
The results shown in Figs. 6.12-6.14 represent the behavior of a system w disturbance. Corresponding results are shown in Figs. 6.15-6.17 for a system is subject to first-order Markov disturbance. The control effort shown in Fig. 6 some effects of plant disturbance, which can be observed by comparing Figs. 6. The reason for this is that the stabilizing feedback for P ( z ) carries disturbance c at the plant output back into the plant input. A plant without this feedback wou same eontrol effort with disturbance or without disturbance. The system being im is like that of Fig. 6.3 or 6.4. For all of the above cases, the command input has been a zero-mea Markov process. It is interesting to challenge the adaptive inverse control system a variable mean to the command input. This was tried with the nonminimumwith k = 24. The reference model M ( z ) was a simple delay of 50 sampling p inverse controller c^ had 100 weights. The convolution of the impulse respons that of G ( z )is shown in Fig. 6.6, and it looks like a perfect impulse delayed by 5 periods. Excellent tracking of the command input is shown in Fig. 6.7, and th prise. However, when a step is added to the unbiased command input, the trac are poor.
-0.02
10
20
30
40
60
70
Time (0.1 sec.)
80
90
Figure 6.15 Convolution of impulse response G ( z ) fork = 24 with impulse response of model verse. The plant was subject to disturbance. The dotted curve is the impulse response of M ( z ) .
- Plant Output ---- Reference Model Output
I
V."
-
-0.3 -0.4
-
I -0.5
20
40
60
80
120
140
160
180
Time (0.1 sec.)
Figure 6.16 Tracking performance: A comparison of the plant output (with plant disturbance sub the reference model output.
-1
2
0
4
0
6
0
8
0
120
140
160
180
Time (0.1 sec.) Figure 6.17
The control effort: The plant input signal.
In order to get a perfect step response, the DC gain from command input t put would need to be unity. The DC gain can be found from the convolution taking the sum of its constituent impulse values. Even though the convolution lo perfect, the sum of the impulses turns out to be 0.68. Tracking is shown in Fig system has r p been trained to deal with this kind of command input. A much response in C is needed. Since the plant is nonminimum-phase, a better coGroller response at zero can be obtained by increasing the number of weights in C , and by increasing th delay. The number of weights was increased to 200, and the delay was increa sampling periods. Checking the sum of the impulses of the new convolution sh be 0.98. The corresponding tracking result is excellent, and this can be seen in This result is an especially good one, since the system still has not been specific to deal with a command input signal containing step functions. It has been tr zero-mean first-order Markov inputs only.
APPLICATION TO REAL-TIME BLOOD PRESSURE CO
A problem was presented by Dr. Mark Yelderman and Dr. William New,’ for the department of anesthesia at the Stanford University Medical Center, concer pressure regulation during postsurgical recovery. Appropriate drug infusion is to prevent the patient’s blood pressure from exceeding a selected limit while a
’Dr. New is a founder of Neilcor Inc. of Hayward, CA. Nellcor manufactures a blood oximeter in gives real-time readings of the percentage of oxygen saturation in the blood.
$
.-a f" d
6
-2
-4
20
10
30
40
6
Time (0.1 sec.)
0
7
0
8
0
9
0
Figure 6.18 Tracking performance: A comparison of the plant output with the command input s by 50 sampling periods, A step was added at time zero to the unbiased command input. C had 100
8
0
7
0
8
0
9
0
110
120
130
140
Time (0.1 sec.)
Figure 6.19 Tracking performance: A comparison of the plant output with the cornman? input s by 100 sampling periods. A step was added at time zero to the unbiased command input. C had 20
shown symbolically in Fig. 6.20. The transfer function and impulse response of tal animals have been observed with this apparatus. A typical impulse respons dog to a 2-mg dose of sodium nitroprusside, a pressure reducing vasodilator dru in Fig. 6.2 1. Notice that the impulse response was negative going, correspondin pressure. The dog behaved like a “black box,” with drug flow as its input and w blood pressure as its output.
Figure 6.20
Pictorial representation of the experimental system. From B . WIDROWand S.D.Ste
signal processing, (Englewood Cliffs, NJ: Prentice Hall, 1985).
An adaptive control system like the one of Fig. 6.4 was simulatedon a labo kuter to control the dog’s blood pressure. A recorded versio? of the dog’s impul Pk(z),obtained by real-time adaptation, was used to find Ck(z). A reference m was chosen to h e e the step response shown in Fig. 6.22. After convergence o computation of Ck(z),the step response of the entire system was as shown in The difference between this step response and the step response of the referen Fig. 6.22 was almost imperceptible. When controlling the recorded impulse response, the simulateddrug flow w to go positive or negative to obtain the result of Fig. 6.23. Control of this type c ically achieved by using a combination of drugs. Notwithstanding this, unidire trol with a single drug is possible, although not as precise. Using the same adap system of Fig. 6.4 except that the simulated drug flow was always positive an
6o
t,
Figure 6.21
5
10
20 30 35 Sample time k ; Sample period = 1 sec.
40
45
Impulse response of dog; 22-kg dog, 2-mg dose of sodium nitroprusside
12c
11c X
E E
2 90
v
; v1
80
5
i
70 60
5
10
20 30 35 Sample time k ; Sample period = 1 sec.
Figure 6.22
40
Step response of reference model.
45
5
Figure 6.23
10
20 30 35 Sample time k; Sample period = 1 sec.
40
45
Step response of controller and dog. Bidirectional driving function.
when the system called for negative flow, the step response shown in Fig. 6.24 w The difference between this response and the model response is visible but slig
I Figure 6.24
5
10
20 30 35 40 Sample time k ; Sample period = 1 sec.
I 45
Step response of controller and dog. Unidirectional driving function.
The adaptive control scheme of Fig. 6.4 appears to be robust and relativ free. Much more work needs to be done, however, to understand its behavior particular to understand how it works in a nonlinear control context. The dynam of the mammalian blood pressure system is certainly not linear, yet ignoring thi sonable results. On the experimental side, a fully automatic blood pressure reg tem is under development as sketched in Fig. 6.20. Using an IBM personal com to the dog during the surgery, several teams of Stanford students working with
SUMMARY
This chapter shows how inverse plant models and model-reference inverse plant be used as controllers in adaptive control systems. An analysis of error is present las are derived for optimal power levels for dither noise used in the plant model and for error power at the plant output. These formulas were checked by expe comparisons between experiment and theory show close agreement. Some of the key formulas are the following: The power of the difference between the plant output and the reference m is given by
The optimal dither is determined by
The next chapter presents other means for achievinkadaptive inverse con when the plant is noisy and usable when there is error in P ( 2 ) .
Bibliography for Chapter 6
[I] B. WIDROW,and J.M. MCCOOL,“A comparison of adaptive algorithm the methods of steepest descent and random search,” IEEE Trans. on An Propag., Vol. AP-24, No. 5 (September 1976), pp. 615-637.
[2] B. WIDROW, J. MCCOOL,and B. MEDOFF,“AdaptiveControl by Inverse in Con$ Rec. of J2th Asilomar Con$ on Circuits, Systems and Computers S CA (November 1978) pp. 90-94.
[3] N. Levinson, “The Wiener RMS (root-mean-square) error criterion in filter prediction,” J. Marh. and Phys., Vol. 25 (1947) pp. 261-278.
for Adaptive Inverse Cont
7.0
INTRODUCTION
A very eEectiv5adaptive inverse control system is the one diagrammed in Fig. 6.4. A
ing that P and C have adequate numbers of weights, this system develops, by adapt controller that when cascaded with the plant provides a very close dynamic match to erence model. However, if truncation effects were significant, the system of Fig. 6.4 develop a controller whose impulse response and transfer function would be biased r to that of the ideal controller, the one that minimizes the mean square of the overall error. It is the purpose of this chapter to introduce new forms of adaptive inverse giv+g inverse controllers that are much less sensitive to truncation and other types o in P. Figure 7.1 is similar to Fig. 6.4, except that the process for finding c^ is online than offline.JThere is not much advantage to this, except perhaps that the modeling for finding C is a natural one. The real reason for introducing the system of Fig. 7 is to provide contrast for the system of Fig. 7.2. The latter is an adaptive inverse system that is significantly different from the system of Fig. 6.4. Figures 7.1 and almost alike, only differing in the source of errorzignal for the adaptation of ?(z). B system of Fig. 7.1 is highly sensitive to errors in P,whereas the system of Fig. 7.2 is insensitive to errors in P. In this chapter, we shall analyze two systems namedfiltered-X andfiltered-6 are capable of adaptive inverse control with low sensitivity to plant model errors.
7.1 THE FILTERED-X LMS ALGORITHM
c(z)
The system of Fig. 7.2 does online2daptation of and uses the overall system e the adaptation process. A copy of C ( z ) is the system controller. This algorithm for ing ?(z) is called the filtered-X LMS algorithm. It is derived in Widrow and Stearn pp. 288-300), and a number of applications are given there. This algorithm gets its from the fact that the input to the adaptive filter providing the X-vector is filtered. references to this algorithm in its various forms are [2,3,4].
Plant disturbance nt
Command input
II
- +
+
COPY C(Z) f
Controller
Dither 8k (white noise)
r 1 -
f I
A
+
* Figure 7.1
Reference model M(Z)
An adaptive inverse control system with online adaptation of e(z).
Plant disturbance nt
Command input
-
Dither 8k (white noise)
I
/
I -
&)
Reference model
,
Wr) Figure 7.2 An adaptive inverse control system with online adaptation of (filtered-X LMS algorithm).
?(I.) using overa
8 k propagated through the plant P ( z ) . Figure 7.3 will be used for the analysis of of Fig. 7.2.
Plant
Figure 7.3
4
Pl distur
A diagram which is useful for the analysis of the system of Fig. 7.2 (filtered-X LMS
Using the same notation as in Fig. 7.3, we will analyzelhe algorithm fo c ( z ) . We will show that this algorithm causes the weights of C(z) to_form the least squares model-reference inverse of the plant P ( z > . The filter C ( z ) conv unusual Wiener solution. The mathematical techniques of Chapter 2 will be use this solution. Suppose that the impulse response of is not constrained. The Wiener obtained as follows. Refer to Fig. 7.3. Recall that the crosscorrelation functio the error and the input signal is zero for a Wiener filgr. Also recall that the ins gradient used by the LMS algorithm when adapting C is proportional to the pro error 6 k and the input signal vector defined Xi.For the system of Fig. 7.3, re continual use of the LMS algorithm to adapt C(z) causes the product of 6 k and Xi small andzventually the expected value of this product goes to zero. Strange as it adapting C(z) with a l%ast squares algorithm does not cause the mean square o 6 k to be minimized if P ( z ) differs from P ( z ) . The reason is that the output of nowhere and is not a part of the error E L . Instead, least squares adaptation simply correlation between 6 k and xi to go to zero. Accordingly, 0 = E[X;ck+ml
This is the Wiener-Hopf equation for the filter ?(z). In getting this result, we as both plant disturbance and dither noise at the plant output are uncorrelated with Equation (7.1) involves a simple convolution and may be easily z-transfo @x'd(Z)
= E(Z)@,k(Z).
From this we obtain
It is useful to compareihis result with the basic Wiener solution of Eq.(2.30). In order to find C ( z ) ,it is necessary to obtain @ x ~ , j (and ~ ) @ x ~ x ( zRef ). again to Fig. 7.3,
Also, from Fig. 7.3,
Therefore, the Wiener solution is
This is the perfect solution! Adapting c ( z )(as in both Figs. 7.2 and 7.3) leads to solution. Note that the value of F(z)has no effect on the Wiener solution. In practice, the perfect solution would generally not be obtained. One re ? ( z ) would not be unconstrained but would be causal and FIR. Another reas perfect solution might not ke achieved is that the adaptive algorithm might no Obtaining convergencefor C ( z )with the system of Fig. 7.1 is s%~ghtforward.C for ? ( z ) with the system of Fig. 7.2 is only straightforwardif P ( z ) = P ( z ) . qth convergence process involves some unk_nowns.The only reason for wanting P t to P itself is to ensure convergence of C ( z ) . To gaii a better understanding of the convergence process, let us apply t gorithm to C(z) of Fig. 7.3. The effects of plant disturbance and dither noise output will drop out, so they are omitted at the outset. We obtain ek+l
=e
k
-k 2p(dk - x:^ck)x;
P’ E[dkx;]. To obtain this result, we have assumed that Xk and Xi are uncorrelated with e k . If (7.8) converges, then A
lim Elek+l] = lim E[ck],
k+m
k+m
and we have
lim E[&J= (R’)-’P.
&-+cr,
I t is evident from (7.8) that convergence of
e in the mean corresponds
to
lim (I - 2pR’Ik = 0.
k-rm
It should be noted that the matrix R is generally not symmetric. It is possi
p(z)could be chosen with sufficient error that the resulting R’matrix, when enter
(7.11). would cause instability regardless oftJhe choice of p. In most cases, howev would not be a problem. Particularly when P ( z ) is close to P ( z ) . Then R’ become to being symmetric and achieving stability would be easy. When u,sing the filtered-X LMS algor@n, it is not necessary to wait foifu vergence of P ( z ) before beginning to acdapt C ( z ) . The rea9on is that a perfect P ( z necessary for the proper adaptation of C(z), as long as the adaptive process is stabl Frgence of the entire system of Fig. 7.2 is generally not much slower than converg P (I) alone. Calculation of the noise in the weights of ? ( z ) is a complicated issue. This toke causes modulation of the command input and in turn causes noise at the ou C ( z ) .This propagates through the plant and causes noise at the plant output. A calc of this noise has been made and the results will be discussed below. Stability conditi the adaptation of e ( z ) of Fig. 7.2 have been obtained and will also be discussed bel
T H E FILTERED+ L MS ALGORITHM
The system of Fig. 7.4 reiresents another approach to fiJding ? ( z ) that does not cr rely on the accuracy of P. The algorithm for finding C ( z ) according to this appr called the filtered+ LMS algorithm. To the best of our knowledge, there is no prior lit for this algorithm. An ideal controller is pictured in Fig. 7.4. If this were the actual controller ins ? copy, the mean square of the overall system error 6 ) would be minimized. The ob
+
Delay A
-
# 6z)
Filtered error error
~~~
Figure 7.4
~~
~~
The filtered-€ LMS algorithm.
is t2make ?I be as close as possible to the ideal C ( z ) . The difference between of C ( z )copy and C(z), both driven by the command input, is thzrefore an error can call this error E'. If E' were available for the adaptation of C(z), the result ? ( z ) that would adapt rapidly and directly toward C ( z ) .The error E' is an ideal for ? ( z ) . The only problem is that it is unavailable. Refer again to Fig. 7.4. Ajltered error, obtained by filtering the overall s ~ k is, used for adaptation in place of E'. The filter is a delayed inverse of P ( z ) verse were perfect and if the plant were free of disturbance, then the filtered erro exactly equivalent to E' for adaptgion purposes. In spite of its limitations, the fi does a fine job for adaptation of C ( z ) . The reasons for this are (a) the plantcJis uncorrelated with the plant input and $e command input, and (b) errors in Pi' critical to the correct convergence2f C ( z ) ,as will be shzwn below. Since a delayed invece of P ( z ) , represented by Pi'(z), is used to filter system error r , the input to C(z) mu2t be correspondinglydelayed so that the inp of ?(z) are time synchronized. As C ( z ) adapts by LMS, its weights are copied in goes nowhere. Only its weights are used by the con troller. The output of Figure 7.5 shows morEdetai1 regarding the implementation of the filtered gorithm. The plant model P ( z ) is obtained in this case by using dither schem plant modeling schzmes could have been used, with or without dither.) An offl utilizes a copy of P ( z ) to obtain its kverse. Generally, a modeling delay will b geous to use, and a delayed inverse Pi'(2) results. There is no performance pe
e(~)
Command input ik I
I
1
Filtered error
Overall system error
+
Reference model M(z)
iNTq+q-j; I
I
I I I I I I
I I I I I I
Delay A
Figure 7.5
I
The filtered-c LMS algorithm with details
the delay A as long as the input to z ( z ) undergoes the same delay. We assum command input ik is stationary. It should be noted that if the plant P ( z ) is minimum-phase, an excellen obtainable without delay A . As such, the filtered error of Fig. 7.5 would not and this error could be used to adapt the controllEr directly. The system could be There would be no need for the separate filters, C ( z )and its copy, only the contro be needed.
have been made by commuting the various filtering operations, permissible with invariant systems.
Plant disturba and dither no llk +8k * P
Figure 7.6
Block diagram equivalent to that of Fig. 7.4.
Comparison of Fig. 7.7 with Fig. 7.3 reveals great similarities between th LMS and filtered-€ LMS algorithms. But there are differences. The input signal in Fig. 7.7 and the noise (the sum of the disturbance at the plant output and t the plant output) is als%filtered. This is not the case in Fig. 7.3. However, the the filters_surroundingC(z), comprised of the cascade of P ( z ) with ?(z), the ca (z) with C ( z ) ,and M ( z ) , and their interconnections, is identical when compari and 7.7.
M(z)
With the filtered-X LMS algorithm (Fig. 7.3). the unconstrained Wiener s e ( z ) was given by (7.6) as
A
This is an ideal solution, and is unaffected by lack of precision in P ( z ) . The de (7.6) made no assumptions about the input ik and about the plant and dither noise the noises are independent of input ik. Although the input and noise are filtered in Fig. 7.7, the same assumptions are valid, and once again the Wiener soluti by (7.12). We can therefore conclude that the filtered+ LMS_algorithm yield solution that is unaffected by choice of P ( z ) and by choice of Pi' (z). Errors i in pi'(z) will affect stability and convergence of the adaptive algorithm but wil the Wiener solution when convergence takes place. This is a pleasing result.
ferent systems having totally different uses, the discovery of similarity in their b a surprising result. Scheme C is described in Chapter 4,diagrammed in Fig. 4. analyzed in some detail in Appendix B. Once again, we shall make block diagrams of the filtered-c LMS algorithm mute operators in the block diagrams. Adaptive filters have time-variant impulse and are normally not commutable with time-invariant linear filters. If we assume that the impulse responses of the adaptive filters are close to their Wiener soluti therefore almost constant, an approximate analysis can be made by allowing co of adaptive filtering (linear with fixed weight vector) with time-invariant linear Accordingly, we begin with the block diagram for filtered-c LMS in Fig. immediately to the equivalent diagram of Fig. 7.7. From there, by a series of s wegofromFig.7.7toFigs.7.8,7.9,7.10,7.11,7.12,andfinallyto7.13 The verify the equivalence between the block diagram of Fig. 7.13 and that of Fig. 7. the steps.
4
I
4
Plant disturbance and dither noise Ilk
Figure 7.8
6k
*P
Block diagram equivalent 10 that of Fig. 7.7.
Although scheme C and the filtered-c LMS algorithm are completely di have different purposes and applications, there is a strong resemblance between
+ Plant and dither disturbance noise nk
Figure 7.9
*p
Block diagram equivalent to that of Fig. 7.8.
Plant disturbance and dither noise Ilk +8k * p
Figure 7.10
A-p
4-p
Block diagram equivalent to that of Fig. 7.9.
Plant disturbance and dither noise Figure 7.11
Block diagram equivalent to that of Fig. 7.10.
Plant disturbance and dither noise nk
Figure 7.12
+6k * P
4-p
Block diagram equivalent to that of Fig. 7.1 I .
Figure 7.13
Block diagram equivalent
10that
of Fig. 7.12.
diagram of Fig. 7.13 and that of scheme C in Fig. 4.3(c). To the right of the do Fig. 7.13 is a block diagram that is structurally identical to the block diagram of In Fig. 7.13, the signal on line A is uncorrelated with that on line B. The this is that the weight noise A P ( z ) in the vector p(z)is random and uncorrelated ponent to component and uncorrelated with the command input ik. The signal o noise, while the signal on line B is a delayed version of the command input. Th is somewhat different in Fig. 4.3(c), with the signal on the line analogous to A controller output, and with the signal on the line analogous to B being indepen noise. The key agreement is that the signals on lines A and B are uncorrelated other in both cases. Furthermore, the noise in scheme C is plant disturbance, an ogous noise in Fig. 7.13 is a combination of plant disturbance and noise at the pl In both cases, these noises are uncorrelated with the signals on lines A and B. Since the input signals and noises have analogous properties and since th of scheme C is analogous to that of filtered-c LMS.we obtain from Appendix B to analyze the filtered-c LMS algorithm. The time constants for filtered-c LMS can be obtained by making use of [The input signal on line B in Fig. 7.13 is not white, so Eq. (B.34) cannot be equation of the same form as (B.32) is Eq. (A.7) whose solution yielded an exp time constants (A. 13a). We use this solution as follows:
where p2 is the LMS adaptation constant for ?(z) withjltered-6 LMS,and A p eigenvalue of the autocorrelation matrix of the input to C(z) in Fig. 7.13 or Fig input is the command input ik.
by (B.53) which, for small p l . reduces to p1 E [ n : ] . The total power on lines therefore Qi3
+ E[(i;)21PlnE[n;l.
From (BSO), the condition for convergence of the variance of the filtered-r LM is
where m is the number of weights in the controller c ( z ) . The weight noise covariance matrix can be gotten from (B.54). For the filt algorithm, weight noise covariance matrix
]
= 1
j~2Cmin I. - 1 2 r n [ ~ [ i , 2 1 +~ [ ( i ; ) * ] p l n ~ [ n : ] ]
For small j ~ z this , becomes weight noise
To get &,,in, refer to Fig. 7.13. This is the power of the noise rk coming from the fi This noise can be written as
The symbols P and pi1represent impulse responses of the correspondingfilter imum mean square error is therefore Cmin
=~
[ r l=l ~ [ 6 , 2 1 +E[(nk * P--I A
2
I.
The dither power is kn_own.The plant disturbance power is generally known, but of the plant inverse P i ' ( z ) would be required to evaluate or estimate the sec (7.19). In any event, weight noise covariance matrix
]
= p2 (E[6,21+ E [ ( n ,
* F 3 1 ) I.
and that ACk (z) changes slowly, commutation is possible and Fig. 7.14(b) also the mechanism generating the dynamic system error. Using Fig. 7.14(b), the var dynamic system error is obtained as variance of = E[(i~)2]/.42m&,,i,.
system error
ik
b
Ack
ik
b
P F
b
-
:i
Dynamic systemerro
P
*
Act
Dynamic system erro
(b) Figure 7.14
Evaluation of dynamic system error with filtered+ LMS.
is given by (7.19). E [ ( i ; ) 2 ]is the power of the output of the plant driven b mand input ik. Some knowledge of plant dynamics is needed to calculate or est Equation (7.21) can be rewritten as emin
variance of system error
+
= E[(ik * ~ ) ~ 1 1 . ~ (2Er [ns , ~ I E[(nk *
PZ;')*]).
This completes our analysis of the filtered-c LMS algorithm and its application t inverse control.
SIMULATION OF A N ADAPTIVE INVERSE CONTROL SYSTEM BASED ON THE FILTERED-€ LMS ALGORITH
In order to verify the effectiveness of the filtered+ LMS and to verify Eqs. (7.1 and (7.22), which describe key behavioral aspects of the algorithm, a simulation o
Various simulation parameters were chosen as follows: a. The command input was a zero-mean white signal of unit power, that is,
b. Dither (used in scheme C) was a zero-mean white noise of unit pow E [ $ ] = 1.
c. The plant noise was zero-mean and white. Its power was a variable param simulation.
d. The number of weights for p(z)was n = 10, and the number of weights fo rn = 10.
These parameters were constant throughout the series of filtered+ expetm variable simulation parameters were the adaptation constant P I ,for adapting P adapting C(z),and the plant noise power E [ n : ] . For each variation of these para simulation was run for a few thousand data samples, about four time constants, steady flow before making measurements. The initial condition for the controller transfer function was c^(z)-= 2, d chosen to be far from the optimal transfer function. The convergence of C(z) on C(z) = I / P ( z ) was noted. The time const_ant of the learning process was obtai serving the changes of the weight vector Ck as it adapted toward the ideal we C. The variance of the weights in steady state was measured, as was the varianc namic system error. Each simulation was run 10 times and the results were avera results are presented in Table 7.1, along with theoretical predictions from (7.13), (7.22). In this case, Eqs. (7.13), (7.16). and (7.22) were able to be utilized quite deed, in order to utilize (7.13), we needed to know only p2 (chosen by the de A,-eigenvalues of the command input's autocorrelation matrix. Since the com was a white zero-mean unit-power signal, all of its eigenvalues were equal to 1 this case, 1 7 = -, 2P2
In order to compute &,in we needed to know the dither power (which was s --I 2 We also needed to know E [ ( n &* PA ) I, which is the output power of the plant were fed with the plant disturbance as an input signal. This can be approximated
TABLE 7.1
PI
0.001 0.0003 0.001 0.001 0.001 0.001 0.001 0.01 0.06 0.1
p2
500
500 1667 500 500 500 500 50 8 unstable unstable
727 2015 655 604 unstable 5 19 unstable 92
Time Constant of c^ Estimated Measured 0.0227 0.00667 0.0227 0.137 1.289 0.0228 0.0228 0.25 3.44
0.0234 0.0053 0.0225 0.177 0.033 1 0.253 3.78 -
Weight Noise Variance for c^ Estimated Measured
0.03 0.009 0.03 0.18 1.68 0.03 0.03 0.3 1.8 -
0.265 4.43 -
-
0.0351 0.0043 0.028 0. I47 0.0307
Dynamic System Error Variance Estimated Measured
EXPERIMENTAL VERIFICATION OF THE FILTERED-
1 1 1
I 10 100 1 1
1
1
E[n:]
Variable Parameters
0 0 0.00 1 0.001 0.001 0.01 0.02 0.001 0.00 1 0.001
Substituting into (7.19), tmin
+
= E[S;J E [ ( Q * pZ;1)2~ 5 = 1 ,-E[n:].
+
Knowing h i n rwe were able to compute the noise variance of the weights of th From (7.16), variance of the noise in the controller weights
1.~2mtmin
- p 2 m [E[i;1 + E ~ ( i ; ) 2 1 ~ l n E [ n' ~ l
411the parameters were known in (7.29) except for ~ ? [ ( i ; ) ~which ] , was the outp
P i ' (z) driven by the command input i k . Since the command input was white, and of unit power, sum of squares of impulses of impulse response of PZ;' ( z )
= 1'
(i)
Now (7.29) was able to be evaluated: variance of the noise in the controller weights
+
lO(1 $[n:J) 1 - p 2 . lO(1 ;wl(lo)E[n;l)' p2
*
+
We wanted to estimate the variance of the dynamic system error, using erything was known except for E[(ik * P ) 2 ] .This was obtained in the followin command input ik was a white, zero-mean, unit power signal. The vector P is o plant impulse response vector. Accordingly, E[(ik * P)'] = 1
- _4 -
3'
sum of squares of impulses of impulse response of plant P
Equations (7.25), (7.3 I), and (7.33) allowed us to predict the time const controller weights, the variance of the controller weight noise, and the variance namic system error for each choice of P I ,p2, and E [ n 3 The theoretical predicti experimental measurements were summarized and compared in Table 7.1. The commentary will help in making an interpretation of the results of Table 7.1.In simulations 1 and 2, the adaptation constant pl for the plant model P ( z constant at the value PI = 0. TJe idea was to demonstrate that the filtered-r algo indeed insensitive to errors in P ( z ) . Fixed errors were deliberately introduced in accord with
F(z)= P ( z ) + 0.4 (1 + Z - ’
+ . + z - ~ ). *
9
If the controller E(z)was computed based on th5erroneous plant model of (7.34 dynamic system error variance (due to error in P ( z ) )would have been quite lar 0.1 19. However, application of filtere_d-E adaptation reduced this variance to m levels. The effect zf the fixed error in P (z) was rendered inconsequential,and onl in the weights of C(z) due to adaptation contributed to the dynamic system erro Instead of this variance being 0.119 for experiment 2, it turned out to be only 0 theoretical prediction was 0.009). Next, we examined the impact of plant disturbance on system performanc lations 3,4, and 5, both pi andp2 were kept constant and the value of .!?[nil was v theoretical time constant for Ck (z) remained unchanged. Experimental time co experiments 3 and 4 were close to theoretical. Weight noise variance and dynam error variance checked well between theory andzxperiment. But simulation 5 ble. What happened was that the plant model P ( z ) was so far away from P ( of such large plant disturbance (E[ni] = loo), instabiJty resulted. The only wa could have been Eevented \odd have been to adapt P ( z ) with much smaller pl attempt to adapt C ( z ) until P ( z ) converged closer to P ( z ) . The parameters p2 and E [ n i ] were kept constant for simulations 3 , 6 , 7, a rameter p~ was varied. As expected from theory, there was no significant chang time constant or weight noise variance or dynamic system error variance. Result and experiment agreed well. But simulation 7 went unstable. In this case, inst predicted from Eq.(7.15). The value of pl was made large enough for instabili Finally, in simulations 3 , 8 , 9 , and 10, we maintained p~ and E[n:] unchE 11.2 was increased from 0.001 to 0.1. Theory predicts that the time constant of C b,” reduced in proportion to increase in p 2 . At the same time, the weight noise v Ck ( z ) and the variance of the dynamic system error would increase with increas
that the stability criterion of (7.15) is quite accurate. The exceptional situation is the case of instability caused by very large p error, due to large plant disturbance or due to a large value P I .Then the conditi is not tight enough to guarantee stability. The reason is that for large errors i tered error ceases to approximate the correct error for adapting the controller. H troller adaptation can no longer be regarded as an approximation of the conven algorithm, and the entire process becomes unstable. While designing a filtere one must keep the plant model error from becoming too large. However, once achieved, the overall system error becomes insensitive to the error in the plant m the entire system is robust and behaves well, as predicted by theory.
,5
EVALUATION A N D SIMULATION OF T H E FILTEREDALGORITHM
A complete mathematical analysis of the adaptive inverse control system ba filtered-X LMS algorithm of Fig. 7.2 is not presented in this book. Although of analysis for the filtered-X is similar to that for the filtered-c algorithm, it is s more complicated. Without going into the details, some results are given next. For the inverse control system based on filteEd-X and using scheme C for eling, the range of 11that allows convergence of P ( z ) is given by (BSO),with E to the plant input signal power and n 3 u a l to the number of weights of P ( z ) . Convergence of the variance of C(z) is obtained if p2 is chosen from withi o
1
+
m[E[(ik * P ) ~ I ~ [ i ~ ] p l n ~ [ n ~ ] ] ’
where m equals number of weights of ?(z), n equals numbekof weights of P ( z adaptation constant of c^(z), p1 is the adaptation constant of P ( z ) , E [ n i ] is the p disturbance power, E [ i i ] is the power of the command input signal, and E[(ik * output power of the plant if it were driven yb- the command input. The time constant for the weights P ( z ) is obtained from (B.34). E [ S i ] i power. The learning time constants for C(z) are given by (7.37), 4
zf
where p2 is the adaptation constant of ?(z), and A p is the pth eigenvalue of the a tion matrix of the output signal from the plant P ( z ) when driven by the input i k .
Cy=,pi2 is the sum of the squares of the weights of P ( z ) or approximately the squares of the weights of F(z),and E [ ( i k * P)'] is the power of the output of th were driven by the command input signal. In order to verify Eqs. (7.36)-(7.38), we have constructed in computer sim system of Fig. 7.2 (filtered-X LMS algorithm). We have chosen to experiment w the same plant, same plant disturbFce, same dither, same control system ymm and same number of weights for P ( z ) , and same number of weights for C(z) scribed in Section 7.4 for the filtered-€system. The plant and reference model ar by (7.23) and (7.24) respectively, and the experimental parameters still hold for simulation. As before, we chose various design parameters to vary: the plant model tation constant P I ,the controller c^(z) adaptation constant pz, and the plant d power E [ n i ] .For eacJ set of parameter values, the system of Fig. 7.2 w e run st the initial value for C ( z ) = 2. Time constants for the convergence of C(z) to obtained. After allowing the system to run for_at least four time constants to allo tem to settle, the variance of weight noise in C(z) and the variance of the dynam error were measured. For each set of parameter values, 10 different simulation and averaged. Results are presented in Table 7.2. TABLE 7.2
EXPERIMENTAL VERIFICATION OF THE FILTERED-X LMS ALGOR
Simulation Number I 2 3 4 5
6 7
8 9 10
fi1
Variable Parameters ~2 ,!?[nil
0
0 0.001 0.001 0.001 0.001 0.01 0.02 0.001 0.001
0.001 0.003 0.001 0.001 0.001 0.001 0.001 0.001 0.01 0.03
Average Time Constant Estimated Measured
I
375
I
1250
I 100
375 375 375
1
375
I I I
38
10
loo0
494 2118 368 403 350 unstable 410 unstable 25 unstable
Dynamic System Estimated M 0.03 10 0.0314 0.1662 2.702
0 0 0 0 2
0.0342
0
0.9142
0
0.0093
Theoretical predictions of system behavior were obtained by makin Eqs. (7.36)-(7.38), and they were compared with the experiment. The theoreti sions were once again easy to use, and they gave predictions that agreed very we experimental results. In utilizing Eq. (7.36), we needed to know E [ ( i k * P)'].
(
variance of dynamic system error
)
= 11.2 .
.
(: + E [ n ; ] ) (1 + lop!E [ n ; l ) .
In applying (7.37), we needed to know the eigenvalues of the autocorrela of the output signal of the plant if it were driven by the command input signal i k . is I 2
1
f
... ...
I
...
I
2
TI
I].
I -
1
The smallest eigenvalue is 0.483, and the largest eigenvalue is 3.577. Hence spread is relatively small. Accordingly, we approximated all the time constants value,
Hence,
From (7.39), (7.40), and (7.43), for each set of values of the parameters E [ n i ] ,we computed the variance of the dynamic system error and the time const and we determined the stable range of 11.2. Results arein Table 7.2. For simulations 1 and 2, we set p1 = 0 and set P ( z ) in accordance with system was not allowed to adapt and correct the erroneous plant model. The idea and demonstrate the insensitivity of the filtered-X algorithm to errors in the plan deed, without utilization of the filtered-X approach (or alternatively the filtered+ if the controller had been computed based on the erroneous plant model, the dyna error variance would have been 0.1 19. However, by using filtered-XL the dyna error variance was much smaller and was independent of the error in P ( z ) . The close correlation between experimental values of the variance and the theoretical dicted by (7.40). Next, we examined the impact of variation in the power of the plant dis system performance. In simulations 3 , 4 , 5 , and 6, 11.1and 11.2 were kept constan was varied. The time constants remained constant. The dynamic system error v creased with increase in E [ n i ] ,as predicted by Eq. (7.38). For very large plant power (in this case, for E [ n : ] = lOOO), the system became unstable. We conclu
increased from 0.001 to 0.03. The time constant of C ( z )decreased and the vari dynamic system error increased with increase of p2, in accord with Eqs. (7.37) For 1 2 = 0.03, the system became unstable. Hence, the stable range for pq was 0 < p2 < 0.03. The theoretical range was, from (7.39), 0 < p2 < 0.075.
The agreement was not perfect, but not bad. For a wide range of pl and 11.2, Eqs. (7.36)-(7.38) generally provide exc diction of the behavior of the filtered-X LMS algorithm. In cases where the adap stants pl and p2 are close to the ends of their respective stable ranges (such as in 9), the dynamic system error variance turns out to be somewhat larger than pred Equation (7.36) provides a good estimate (within a factor of 2) of the stab 1 1 2 . However, similarly to what we saw with the filtered-c LMS algorithm, when the plant mo*l becomes very large (either due to large plant disturbance power plant model P ( z ) adaptation constant PI), the filtered-X system ceases to appro LMS convergence process. As a result, the entire system becomes unstable (eve chosen well within the range determined by (7.36)). However, for small enoug p(z),the system is stable, robust, and completely insensitive to errors in the pla By comparing Tables 7.1 and 7.2, we conclude that the filtered-X and filte rithms performed similarly to one another. For a given level of dynamic system ance, the filtered-X algorithm provided somewhat smaller time constants on av filtered-c algorithm had only a single time constant. Regarding the experiments the tables, the time constant of the filtered-t algorithm was half that of the slowe the filtered-X algurithm, but it was larger than that of the fastest mode of filtered-X physical plants tend to be low-pass, there generally will be substantial eigenvalue the matrix R I . Under worst case conditions, filtered-X can be slower than filtere ever, on average, filtered-X will be somewhat faster than filtered-c. These conclusions are intuitive and of a rule of thumb nature. In each spe both algorithms should be considered. They should be analyzed both theoretica computer simulation before the best configuration is chosen.
A PRACTICAL EXAMPLE: ADAPTIVE INVERSE CONT FOR N0IS E-C A NCELl N G EAR PH O NES
It is difficult to communicate with a person working in a high-noise environmen ation is helped if the person is wearing earphones and communication is done by
If the fixed filter is designed properly to have the right gain and phase shift as of frequency, a significant noise reduction is possible. Commercial versions of exist, with the fixed filter being an amplifier with a small gain and a small delay
Figure 7.15
A nonadaptive noise-canceling earphone.
The limitations of the scheme of Fig. 7.15 result from variations in peopl ear shapes and variations in how the earphones are worn. There are also ma imperfections that result in variations in the microphone and earphone charact design of the fixed filter involves many compromises. Better results can be obtained with adaptive techniques. The basic idea i in Fig. 7.16. Two microphones are used in the pictured system, one on the oute earphone and the other on the inner face, close to the ear canal. The signals ob these microphones are labeled M Iand M2,respectively. Microphone number o the strong ambient noise (SAN). Microphone number two receives the signal ear. Both microphone signals are input to an adaptive noise canceling system input signal containing the useful audio information goes through this system earphone signal (EPS),but ambient noise components are adaptively filtered an from it. The goal is to reduce and minimize (in the mean square sense) ambien ponents present at the entrance of the ear canal and at the same time to allow the signal to exist there. Figure 7.17 describes the physical situation. Strong ambient noise is rece crophone number one, a transducer whose transfer function is represented by
Figure 7.16
An adaptive noise-canceling earphone
and whose output is represented by the signal M I .The ambient noise propaga the structure of the earphone and resonates in the cavity between the earphone a The resulting acoustic signal is transduced by microphone number two. The tra tion between the ambient noise and the signal M2 is represented by [M2/SAN]. signal component is sensed by microphone number two, and that is caused by th signal (EPS) electrically actuating the loudspeaker in the earphone.' The transf between the earphone signal and the output of microphone number two is des [M2/EPS]. In Fig. 7.17, the formation of signal M2 is therefore represented as a ambient noise components and the earphone signal components. The earphone s equals the audio input signal minus the ambie_nt noise signal after adaptive filt corresponding adaptive filter is designated as C , analogous to the controller of a inverse control system.The weights of C are to be adjusted to minimize the mean square of M2 taken by the adaptive process to be the overall system error. The adaptive proce the power of only the ambient noise components in M2. The audio input comp uncorrelated with the noise, and they add to the power of M2. The adaptive pro these components unaffected. An entire noise canceling system, based on the filtered-c LMS algorithm, i Fig. 7.18. The error M2 could Eot be used directly as the error signal for adapta adaptive noise canceling-filter C . Filtering the error is necessary because the tra tion from the output of C back to M2 is not a simple gain of minus one.2 Instea is -[M2/EPS]. Therefore, it is necessary to model this gain and do a filtered-c
'Because of physical isolation, we assume that the level of earphone signal (EPS) received by number one is negligible.
'If this gain were - 1 , then the adaptive system would be a simple adaptive noise canceler as Widrow, and colleagues [ 5 ] , and by Widrow and Steams [ 11.
Figure 7.17
Some details of adaptive noise-canceling earphone.
In Fig. 7.18, the audio input signal is used in modeling [M2/EPS]. We can re results of the modeling as
F A [M2ZPS].
The filtered P^ has the audio input signal as its input. The microphonesignal M2 i responseAThedelayediyerse of P is obtained offline. This inverse is used as th to adapt C. A copy of C does the actual ambient noise filtering for the cancelin Using the methodology of Fig. 7.18, it should be possible to devise a mo able earphone that would not need to fit as tightly as present earphones do and c porate an air “breathing space” between the ear and the earphone. This would airplane earphone ever, but it would be expensive.
7.7
A N EXAMPLE OF FILTERED-X INVERSE CONTROL OF A MINIMUM-PHASE PLANT
The stabilized minimum-phase plant that was studied previously in Chapter 1 ha trolled by means of the filtered-X algorithm. Figure 7.19 is a block diagram s plant and its stabilization loop driven by a zero-order-holdDAC. The stabilize
trong ent noise
SAW
(AIS)
Audio put signal
F i p m 7.18
Complete details of adaptive noisecanceling earphone.
The stabilized plant and the DAC wer5adaptively modeled, as represented At a given moment of time, the weights of G ( z ) were recorded and then plotted is shown in Fig. 7.23. This is a very ngsy version of the true theoretical impulse Fig. 7.20. The noise in the weights of G(zJ is caused by rapid adaptation, and is i the plant disturbance. Using this noisy G ( z ) with thecltered-X algorithm to o the result is plotted in Fig. 7.24. The convolution of C ( z ) with the true impul G ( z ) is shown in Fig. 7.25. This is a remarkably good result, almost a perfect u with a unit delay, and it compares very well with-the convolution shown in Fig controller ?(z) is almost unaffected by noise in G ( z ) . That is one of the beaut of the fi1tered;X algorithm. Using C ( z ) as a controller, as in Fig. 7.19, excellent tracking of the com signal took place. Figure 7.26 shows superposedplots of the command input sign by one sample time, and the plant output, with the plant disturbance subtracte almost no difference. Figure 7.27 shows superposed plots of the command in delayed by one sample time, and the actual plant output, including the plant d The difference is almost completely due to the plant disturbance. Canceling this will be the subject of the next chapter.
SOME PROBLEMS IN DOING INVERSE CONTROL W I T FILTERED-X LMS ALGORITHM
The filtered-X LMS algorithm requires that the input signal to ?(z) be filtered by the plant. Above, we have studied the u~ of an unstable p@nt being stabilized by loo& shown in Fig. 7.19. The input to C(z)_is filtered by G ( z )which is low-pas by G ( z )causes the spectrum of the input to C ( z )to be far from white. The covari of the inputs to the weights of ?(z) has a wide eigenvalue spread. In some cases o the eigenvalue spread could be so great that convergence with the steepest de algorithm would be impossible to obtain in a practical length of time. of The magnitude of the frequency response of the minimum-phase shown in Fig. 7.28. Although this passes all frequencies, it yields an input sig having an eigenvalue spread of 30,OOO to l.3 Convergencewas slow but sure in th the other hand, when doing the filtered-X LMS algorithm with the nonminimum(stabilized as in Fig. 1.17 with k = 24, and with M ( z ) chosen to be a 50 time
c(z)
3Thespectrum of the input to a transversal filter determines the eigenvalue spread of the R-matri the largest to the smallest Gigenvalue equals the ratio of the peak value of the spectral density to th of the spectral density [6,7].
2I
4
Controller
S
1 -z-' -
k=4
Referencemodel M(z)
,
Plant
Filtered-X control of stabilized minimum-phase plant.
O-order hold DAC
Figure 7.19
8a .-
-
a
0.2
Y
3
0.15
0.1
0.05
Time (0.1 sec.)
Figure 7.u) Theoretical impulse response of G(z), minimum-phase plant of Fig. 7.1
I
5
10
20
30
Time (0.1 sec.) Figure 7.21
35
40
Theoretical unit-delay inverse of G(z).
45
-0.2'I
5
10
20
'--
30
Time (0.1 sec.) Figure 7.22
35
I
40
45
Convolution of theoretical G(z) with theoretical unit-delay inverse.
-. . -
150
Time (0.1 sec.) Figure 7.23
A noisy G^(z)
H.-Y
8 4
0.
-0.
-1 .!
-
5
20
10
3
0
3
5
4
0
4
5
5
4
0
4
5
Time (0.1 sec.) Figure 7.24
e ( z ) obtained with noisy &).
0
0
3.--
E 4 Y
0.
0.
-0.
#
5
10
20
3
0
3
Time (0.1 sec.) Figure 7.25
Convolution of e ( z ) with the true G ( z ) .
8
-
.-an
3
-o.2 -0.4
-0.6 -0.8
5
10
20
3
0
3
5
4
0
4
5
I
Time (0.1 sec.) Figure 7.26
Tracking performance of filtered-X inverse control of stabilized minimum-pha
- Command Input _ _ - _Plant Output
5
10
20
30
Time (0.1 sec.) Figure 7.27
35
40
45
Command input signal, delayed by one sample period and plant output including plant
filtered-X LMS algorithm would not be an appropriatechoice. It takes too long t
>5 ' 2
150
200
Frequency (100 units = 112 sampling freq.) Figure 7.28
Magnitude of transfer function of E ( E )of stabiiized minimum-phase pla
INVERSE CONTROL WITH THE FILTERED-X ALGOR BASED ON DCT/LMS
To get improved convergence when the R-matrix of the input to the adaptive very large eigenvalue spread, algorithms other than LMS are often used. Ple Appendix E for a description of the RLS and the DCT/LMS algorithms which a for this purpose. An attempt was made to use the RLS algorithm to do filtered-X, and this w cessful. We do not know yet why it did not work. However, the DCTLMS alg tried and this was very successful. There was a dramatic improvement in the speed of convergence of ? minimum-phaseplant described in Section 7.8 above. When using the LMS alg eigenvalue spread of the inputs to the weights was 30,000 to 1. When using the algorithm, however, the eigenvalue spread of the inputs to the weights was onl
Frequency (100 units = 1/2 sampling freq.) Figure 7.29
Magnitude of transfer function of &z) of stabilized nonminimum-phase p
Time (0.1 sec.) Figure 7.30
c ( z ) for the stabilized nonminimum-phase plant with k = 24.
Figure 7.31 gorithm.
for the stabilized nonrninirnurn-phase plant, k = 24, after lo6 iterations of the
&)
0.5 7
0.4 -
-
0.3
33 ._ 4-
E 4
0.20.1 -
I
the impulse response of G ( z ) g a t was obtained by DCTLMS adaptation. This used with filtered-X to obtain C(z), shown in Fig. 7.35. Adaptation of c ( z ) was done by DCTLMS. Convolvingthe impulse respon with that of the true G ( z )yields the result shown in Fig. 7.36. This looks like pulse, delayed by 50 sampling periods, matching M(z) which was set to be a pur closer study of the convolution shows that the impulseJas side lobes, and that its is about 0.8. The reason for the imperfections is that C ( z ) was not adapted long To get an almost perfect match to M(z) over a wide range of frequencieswo more adaptation to make the convolution more like a unit impulse. But the comm signal was first-order Markov and low-pass. So, although the convolution was no impulse, excellent tracking of the command input still took place. The system make the error small for the given command input signal. A sample of plant output including the plant disturbance is compared in Fig the same plant output displayed with the disturbance removed. The plots indica plant disturbance was substantial. Figure 7.38 shows the command input, delaye with M(z), and the plant output plotted with the disturbance removed. It can b the tracking is very good indeed, almost perfect.
INVERSE CONTROL WITH T H E FILTERED-€ ALGORI BASED ON D C T / L M S
The filtered-€ algorithm was used to control the stabilized nonminimum-phas was studied previously, in Chapter 1. Figure 7.39 is a block diagram of the sy input to C ( z ) ,as shown in the block diagram, is the command input itself, not filte the filtered-X algorithm, except for a time delay of A. The command input was Markov. The eigenvalue spread was much less severe than with the filtered-X “only” 340 to 1. To speed up the convergenceprocess, LMS was first replaced wi this did not work. The DCTEMS algorithm worked very well and was used in t E experiments for adapting C(z). The eigenvalue spread of the inputs to the w 2 to 1. Figure 7.40 is the impulse response of g(z1obtained by LMS. Figure 7.4 pulse response of G’,I (z), the delayed inverse of G ( z ) ,which could have been ob LMS by the offline process of Fig. 7.39 but-was instead obtained offline with L Figure 7.42 shows the impulse response of C ( z ) ,obtained by filtered+. M ( z ) w to be a simple delay. Figure 7.43 shows the convolution of the impulse respon with that of the true G ( z ) .The convolution is not a perfect delayed impulse, bu enough to allow excellent input-output tracking. Figure 7.44 shows the output o
1 -
1
Controller
Figure 7.33
S
I-z-' -
. DAC
I
#
s+ 1
(s+7)(s-2)
Reference model M(z)=z-W
*
*
/
I-
disturbance
Error
Filtered-X control of stabilized nonminimum-phase plant using the.DCT/LMS algorithm.
-
(
+
Plant output
-"."
Figure 7.34
10
20
30
40
6 0 7 0 8 0 9 0
Time (0.1 sec.)
Impulse response of 6(z)for the stabilized nonminimum-phase plant.
10
20
-
30
40
60
Time (0.1 sec.)
70
80
90
I
Figure 7.35 Impulse response of C ( z )for the stabilized nonminimum-phase plant, obtained by fil DCTLMS adaptation.
,;:
,:r i
----
Disturbance-Free Plant Output Plant Output (with Disturbance)
:: : :
-0Sc
2
0
4
0
6
0
8
0
120
140
160
180
Time (0.1 see.) Figure 7.37
Plant output, and plant output with plant disturbance removed
-0.4
2
0
4
0
6
0
8
0
120
140
160
180
Time (0.1 sec.) Figure 7.38
Tracking performance: Command input signal and plant output with disturbance
including the plant disturbance, and the same output of the plant plotted with the d removed. From this one can see that the disturbance was substantial. Figure 7.45 command input, delayed to correspond to M(z), and the plant output plotted wi disturbance removed. It can be seen that not perfect but very good tracking is ta
SUMMARY
We conclude this chapter by comparing the performance characteristics of adapt control systems based on the filtered-X and filtered-c LMS algorithms with the Fig. 6.4. The system of Fig. 6.4is constructed so that F(z)has enough weighLs, tha enough toke able to model P with a negligible truncation error. Error in P ( z ) w choice of C ( z ) . Tke filtered-c T d filtered-X algorithms, on the other hand, w biased choices of C(z) even if P(z) has some truncation error. They both cau converge in the mean to the ideal controller, the_onethat minimizes the mean sq overall system error in Fig. 6.4.But error in P ( z ) could cause instability in th process that yields ?(z). This does not happen with the system of Fig. 6.4. When designing atadaptive inverse control system, one could follow the a Fig. 6.4and use a long P ( z ) . C_ontrollingstability would be easy, but to Educe of weight noise, adaptation of P(z) would need to be slow. Adaptation of C(z), io at high speed, would-not slow down the overall adaptive process. Therefore, C converge as fast as P (z). Another possibility would be to use the filtered-€ algorithm. Now-p(z) be so long, and its speed of convergence could be allowed to go higher. P(z) w to be long enough and close enough to P ( z ) to enable the algorithm computing
1 -
I
4 I -z-' -
d
S
COPY C(Z) I
O-order hold DAC
I
A 6Z)
f PWR Norm
k = 24
ALin Comb
(S
S+
1
+7)(S - 2)
/
Compensating network
+ Filtered error
Plant
I c_"PY
1
Plant disturbance I
Filtered+ control of stabilized nonminimum-phase plant using the JXWLMS algorithm.
M(z) = 2-m
Reference model
-
5:
Controller
b
~
Delay
Figure 7.39
+
Plant output
-0.1
-
-0.2-0.3L
Figure 7.40
10
W1 30
20
Impulse response of
10
Figure 7.41
20
40
8
0
7
0
8
0
9
0
Time (0.1 sec.)
z(z),nonrninimurn-phase plant with feedback stabiliza
30
40
80
70
80
Time (0.1 sec.)
Impulse response of C ( z ) i ' ,the delayed inverse of
90
c(z).
10
Figure 7.42 delay.
Impulse response of
0.9~I 0.8
-
0.7
-
0.6
3._ 8 u
O5 0.4-
0.3-
0.20.1 -
20
30
40
6 0 7 0 6 0 9 0
Time (0.1 sec.)
e(z)for nonminimum-phaseplant obtained by LXTLMS. M ( z
8 .-a
-a $
-0.2 -0.4
-0.6 -0.8 -1
2
0
4
0
8
0
8
0
120
140
180
180
Time (0.1 sec.) Figure 7.44 removed.
The plant output including its disturbance, and the same plant output plotted with th
0.6
----
J ::
-
Delayed Command Input Disturbance-FreePlant Output
-
:I
-1
2
0
4
0
8
0
8
0
120
140
180
180
Time (0.1 sec.) Figure 7.45 Tracking performance: Comparing delayed command input signal with plant output plant disturbance removed.
P
(2). The learning time constant of
the weights of P(z) are given by (B.34) a
where E[6,2] is the dither power. The variance of the dynamic system error at th which, in turn, is due to the noise of p u s u e to the noise of the weights of in P(z) is given by
e(z)
variance of dynamic system error
)=
+~S(O).
min(rn, n) B ,
where $ 1 ~ ~ is( the 0 ) power output of the controller ?(z), m is the number of wei and n is the number of weights of p(z),min(rn, n) is the smaller number of t equals I the product of p and ,?[nil, the plant output disturbance powe where # For the adaptive inverse control system based on the filtered-6 LMS aigori in Fig. 7.5, the range of P I that allows convergence of the variance of P(z) (B.50). Onse again, E [ u : ] is the power of the pknt input signal and n is the weights of P(z). Convergence of the variance of C(z) is assured if ~2 is chose range
where tn is the number of weights of ?(z), n is the number of weights of p(z) plant output disturbance power,E[iz] is the power of the system command inpu is the output power of a filter P i ’ ( 2 ) driven by the command input i k . The filt the delayed plant inverse which is obtained by an offline adaptive process. Stab simple adaptivcprocess is easy to obtain by making use of (A.36). The time c the wei.ghts of P(z)^is obtained from (B.34), and once again E[S,2]is the dither time constants for C ( t )are given by (7.13).
where j ~ is2 the adaptationzonstant for ?(z) and A,, is the pth eigenvalue of the a tion matrix of the input to C(z),that is, the command input i k . The variance of t system error is given by (7.22) as variance of system error
+
= E[(ik * P)2]~2rn[E[6:l E[(nk * Fi’)2]],
Without going into the details, some results are presented in Section 7.5 and s here. For the adytive inverse control system based on filtered-X, the range of p convergenceof P ( z ) is given by (B.59, with E[u;] equal to the plant input sgna n equal to the number of weights of P ( z ) . Convergence of the variance of C(z) if p2 is chosen from within the range
A
where m equals number of weights of c^(z), n equals numbe5of weights of P ( z adaptation constant of c^(z), p1 is the adaptation constant of P ( z ) , E [ n ; ] is the p plant disturbance, E [ i i ] is the power of the command input signal, and E[(ik * output power of the plant if it were driven by the command input. The time cons weights of p(z)is obtained from (B.34), and E [ $ ] is the dither power. The tim for C are given by
where ~2 is the adaptation constant of I?(z), and k p is the pth eigenvalue of t relation matrix of the output signal from the plant P ( z ) if it were driven by th input i k . The variance of the dynamic system error is given by n
variance of = E[(ik
system error
* P)21p~m[E[6il
pi’
+ E [ n i l ] (1 + npl E [ n
i=I A
where k l a$ p2 are adaptation constants of p(z)and C (z), n and m are the weights of P ( z ) and c^ (z), E[n;] is the power of the plant disturbance at the p E [ S i ] is the dither power, Cblp’ is the sum of the weights of P ( z ) or approx sum of the weights of p(z),and E[(ik * P)’]is the power of the output of the pla driven by the command input signal. Stability,time constants, and dynamic system error can be determined for verse control systems by making use of the equations of this summary. Most of ters of the formulas are readily available to the system designer. Some need to b Exact values will not always be available. The best way to choose from these would be to analyze all three, with the formulas and by computer simulation. When using either the filEred-c or the filtered-X algorithm for adaptive i trol, errors in the plant model P ( z ) , designated by A P ( z ) ,have no effect upon
Bibliography for Chapter 7
[ l ] B. WIDROWand S.D. STEARNS, Adaptive signal processing (Englewood Prentice Hall, 1985).
[2] J.C. BURGESS,“Active adaptive sound control in a duct: A computer s J. Acoust. SOC.Am., Vol. 70, No. 3 (September 1981),pp. 715-726.
[3] B. WIDROW, D. SCHUR,and S. SHAFFER,“On adaptive inverse c Cont Rec. of 15th Asilomar Con6 on Circuits, Systems and Computers, S CA (November 1981), pp. 185-189.
[4] B. WIDROW,“Adaptive inverse control,” in Proc. of the 2ndIFAC Worksho tive Systems in Control and Signal Processing (Lund, Sweden, July 1986), per, Pergamon Press pp. 1-5.
[5] B. WIDROW,J.M. MCCOOL, J.R. GLOVER,JR., J. KAUNITZ,C. W R.H. HEARN,J.R. ZEIDLER,E. DONG, JR., and R.C. GOODLIN,“Ada cancelling: Principles and applications,” Proc. IEEE, Vol. 63, No. 12 (Decem pp. 1692-1716.
[6] U. GRENANDER and G. SZEGO, Toeplitz forms and their applications (Be University of California Press, 1958). [7] R.M. GRAY,“Toeplitz and circulant matrices: 11,” Information Systems Technical Report No. 6504-1, Stanford University, April 1977.
[8] Dozens of applications of the Filtered-X LMS algorithm are given in the P Con& on Recent Advances in Active Control of Soundand Wbration, Virgini nic Institute, April 1991 (Lancaster, PA: Technomics Publishing Co.), Fax: 4538.
8.0
INTRODUCTION
Methods for controlling plant dynamics have been described in the previous chapters. methods have no effect on plant disturbance, which would simply appear unimpeded plant output. A feedback scheme for plant disturbance canceling which does not alte dynamics is suggested in Figs. 1.4 and 1.5. The purpose of this chapter is to develo disturbance canceling techniques based on this scheme. The goal is to minimize plant disturbance power without changing plant dynamics. In control theory, it is most common to control plant response and plant distu in one process. With adaptive inverse control, however, it is convenient to treat these lems independently. In this way, the dynamic control process is not compromised need to reduce plant disturbance. Furthermore, the plant disturbance reduction pro not compromised by the needs of dynamic control. A plant disturbance canceling system is diagrammed in Fig. 8.1. Dizturbance ca tion is accomplished by this system in the following manner. A copy of Pk(z),a very disturbance-free match' to P ( z ) , is fed the same input as the plant P ( z ) . TJe diff between the disturbed output of the plant and the disturbance-freeoutput of Pk ( 2 ) is close approximation to the plant output disturbance nk. TheEpproximate nk is then in the filter z - ' Q k ( z ) which is a best least squares inverse of Pk(z). The output of z - ' is subtracted from the plant input to effect cancelation of the plant disturbance. Unit z-I were placed in front of Qk(z)'s in recognition of the fact that digital feedback link have at least one unit of delay around each 100p.~Thus, the current value of the pla turbance nk can be used only for the cancelation of future values of plant disturban cannot be used for instantaneous self-cancelation. The effects of these unit delays are when the system is operated with a high sampling rate, however.
'
& ( z ) is "disturbance free" in the sense that plant disturbance nk is absent. Noise in the weights of & ( z adaptation will of course exist. However, noise in the weights can be made small by making p small (i.e., adaptation slow). 'In physical systems where the plant is analog, if the plant has more poles in the s-plane than zeros, , have at least a unit delay in its response to an impulse input. A cretized form of the plant, k ' ( ~ ) would the unit delay in line with Qk(z) would be. unnecessary and should be left out. Including it would cause performance.
Figure 8.1
nk
Plant disturbance
c
+
Plant output
An adaptive system for canceling plant disturbance.
reference from the plant output and uses it for canceling byfeedbackfiltering and from the plant input. It tries to cancel out its own disturbance reference signal. N this happens with the conventional adaptive noise canceler. The adaptive plant d canceler represents a wholly new concept in noise canceling for a completely dif of noise (disturbance) control problem.
8.1 T H E FUNCTIONING OF T H E ADAPTIVE PLANT DISTURBANCE CANCELER
Exactly how the system of Fig. 8.1 cancels plant dkturbance can be explained alg Suppose that the system has converged and that P k ( z ) and Q k ( z ) are now fixed to Fig. 8.1, it is useful to note that the transfer function from the input at point A output at point C is
E(z)
= P ( z ) , the transfer fun From this transfer function, one can see that when point A to point C is equal to the transfer function P ( z ) of the plant itself. Th ci-rcuit of the plant disturbance canceler therefore does not change the plant dyna Pk(z) matches P ( z ) , regargess of the value of Q k ( z ) . Furthermore, when P k ( z ) matches P ( z ) , the stability of Eq. (82)'is assu that P ( z ) itself has been assumed to be stable.) On the other hand, if Pk(z) does P ( z ) , then values of Q k ( z ) can easily be found to make the disturbance cance unstable. It is most important to have converged and sustained close to P ( z ) b ing on the disturbance canceling feedback. While running, if therejs a sudden c change in P ( z ) , it may be necessary to shut off the feedback until Pk(z) can hav tunity to lock back onto P ( z ) . A panic button switch should be included in th path in case of instability. It would be better to suffer the effects of plant distur while rather than go unstable. Our next interest is with the transfer function from the point of injection o turbance nk, point B, to the plant output, point c . It can easily be shown to be
g(z)
HBC(Z)
=
1-
z-'E(z)Q ~ ( z ) *
+ z - ' P ( z ) . Q ~ ( z -) z - ' P ~ ( z ) Q ~ ( z ) A
1
*
point D. Minimizing the power at point D, as is done by the adaptive proces Q k ( z ) , thereby causes the power of the plant disturbance component at point D imized. This corresponds to the power of the plant disturbance at the plant ou also being minimized. Therefore, the adaptive process finds the best value of copied and used in the disturbance canceling feedback process for the purpose ing plant output disturbance. The only question that remains is whether or no configuration shown in Fig. 8.1 is optimal. Could some other system arrang another adaptive process to do an even better job of disturbance cancelation?
8.2
PROOF OF OPTIMALITY FOR THE ADAPTIVE PLAN DISTURBANCE CANCELER
Fig. 8.2(a) shows the plant with additive output disturbance nk. The plant ou bance is assumed to have been generated by an original disturbance lk, which of unit power, acting as a driving function input to a linear filter whose transfe r(z).This disturbance model is illustrated in Fig. 8.2(b). The filter r(z)must order to have finite plant output disturbance, Without loss of generality, this f assumed to be stable, minimum-phase, and causal. A plant output disturbanc any desired power spectrum can be generated by the application of a suitably c disturbance source lk to a stable, causal, minimum-phase filter r ( z ) . It is useful the reciprocal of r ( z ) is also a causal, stable filter, and it is a whitening filter fo For control purposes, the only access that one has to the disturbed plan port. In order to reduce the plant output disturbance, one can only add distur plant input. This input disturbance must be properly related to the source lk of th put disturbance in order to effect output disturbance reduction. If the disturban system is to be a linear one, a theoretically optimal plant disturbance canceler figured as shown in Fig. 8.2(c), with F * ( z ) optimized to minimize plant disturb at point F. Subsequently,we shall demonstrate that the behavior of this optimal canceler is equivalent to that of the disturbance canceler of Fig. 8.1. One cannot generally implement the optimal scheme of Fig. 8.2(c). The re one would not usually have access to the source noise lk. An equivalent schem in Fig. 8.2(d) which uses the actual plant output disturbance, not the source no Assuming for the moment that one did have access to the source noise as in the source noise could be filtered3and applied to the plant input to do the best p
'In Figs. 8.2(c) and (d), we actually assume access to rk-1 and m-1, respectively. Delays of zin the noise flow paths because the current value of the plant noise can only be used for the cancel
lk
White noise, unit power
4 I ry)
Stable, causal minimum-phase filter
Causal
Wiener filter (C)
Figure 8.2
Comparative study of optimal and practical plant disturbance canceling schem
tion, the question is, what can one do in reality? The noise canceling system of uses the actual plant output disturbance rather than the unavailable source noi of this system will lead us to a practical result. The noise canceling scheme of Fig. 8.2(d) turns out to be identical in fun optimal canceler of Fig. 8.2(c). This can be demonstrated as follows. Refer 8.2(d) and using the Shannon-Bode method once again, the optimal causal form that minimizes plant disturbance power at point C is
Using (8.4) and (8.51, we can relate - Q * ( z ) and F * ( z ) :
Referring to Fig. 8.2(c), it is apparent that the z-transform of the plant dis point F is L ( z ) [ r ( z ) z - ' F * ( z ) . P (z)l,
+
where L ( z ) is the z-transform of the noise l k . Now referring to Fig. 8.2(d), it that the z-transform of the plant disturbance at point C is
= uz)[r(z)+ z - ' F * ( z ) P ( z ) ] ,
where N ( z ) is the z-transform of the plant disturbance nk. We note that the plant at point C, given by (8.8) is identical to the plant disturbance at point F, given by conclusion is that the Wiener canceler of Fig. 8.2(d) performs equivalently to canceler of Fig. 8.2(c). They are both optimal linear least squares noise canceler linear canceler can give better performance. We'll show next that the disturbance canceler of F i g 8.1 is equivalent to 8.2(d) and is thereby an optimal canceler, assuming that pk(z) = P ( 2 ) . The ad cess for finding Qk ( z ) of Fig. 8.1 is driven by the plant disturbance nk. The s for the Wiener canceler of Fig. 8.2(d). The optimal Q * ( z ) in Fig. 8.2(d) is the values of plant noise. If the discretized plant P ( z ) has at least one unit of delay in its impulse respo of z-' can and should be eliminated.
= L(z)[r(z)
+ z-' w .F*(z)I.
This is the same result as (8.7). Therefore, when P^(z) = P ( z ) , the disturbance Fig. 8.1 performs equivalently to that of Fig. 8.2(d) (which is optimal).
8.3
POWER OF UNCANCELED PLANT DISTURBANCE
Even the best disturbance canceler cannot cancel perfectly and will leave a res canceled plant disturbance. This occurs because the minimum mean square e Wiener filter is generally not zero. An expression for the power of the uncan disturbance can be derived from (8.7) as follows. Since L ( z )corresponds to wh unit power, the transform of the autocorrelation of the uncanceled disturbance output (at point C, Fig. 8.1) will be
+
+
mcc(z) = (r(z) P ( z ) 4 F*(z))(r(z-') ~ ( z - 9. z - ' F * ( z - ' ) ) =~~~+z4cc~-1~+4cc~o~+z-'4cc~
The uncanceled plant output disturbance power will be 4cc (0). This can often b when some knowledge of P ( z ) and r(z)is available. One can factor the plant d spectrum @,i,,(z) to obtain T(z), in accord with @,,m = rww-'). Furthermore, F * ( z ) can be obtained from (8.4)with knowledge of P ( z ) and all of these ingredients, ~ c c ( O can ) be obtained by making use of (8.10).
r(
OFFLINE COMPUTATION OF Q ~ ( z )
If Qk ( z ) were computed by an adaptive process as in Fig. 8.1, very slow adapta be required in order for the effects of misadjustment due to noise in the adapt of Qk ( z ) to be negligiye. On the other hand, fast adaptation might be required an inverse of 6.z) if Pk(z) were tracking a time variable plant P (2). If this we it would be a good idea to calculate Qk(z) by an offline process as illustrated One would need synthetic noise whose statistical character would be the same a original plant disturbance. The offline process could operate much faster than r 'Hundreds or possibly thousands of iterations could be undergone offline in the calculation of of one sampling period of the physical system
Qk
nk
1
SIMULTANEOUS PLANT MODELING AND PLANT DISTURBANCE CANCELING
A plant modeling and plant disturbance canceling system is shown in Fig. 8.4 tem incorporates the plant disturbance canceling scheme of Fig. 8.1 and the plan scheme of Fig. 4.1. An offline inverse modeling p_rocessto obtain Q k ( z ) is show pose of this system is clearly twofold: to obtain f k ( z ) , and to cancel the @ant as well as possible in the least squares sense. Copies of the plant model fk (z) for the disturbance canceler and for the offline inverse modeling process to get
b outp
/
# COPY
Z-I
QtW
~ I
~
Offline process for generating
Figure 8.4
QP (2)
~
[a
i_ -
WARN do not u
~
I
A plant modeling and disturbance-canceling system that does not work.
It is clear that the system of Fig. 8.4 is a straightforward combination of robust subsystems. This entire system was built in software and tested in our and it did not w_ork! Much to our surprise, it was discovered that the system w flawed. When Pk(z) converged, it did not converge on P ( z ) , that is, it did not best match to P ( z ) . Instead it developed a bias. We soon learned to be very ca building "simple systems." In connection with the basic process of Fig. 4.1 for modeling a disturbed (4.6)-(4.13) show that the Wiener solution for Pk(z) will be a best match to P ( of the presence of plant disturbance nk as long as this plant disturbance is uncorre the plant input. But in the system of Fig. 8.4 the plant disturbance is fed back via purpos~sof disturbance cancelation. The result is that the input to the adaptive forms & ( z ) has components related to and correlated with the plant disturbance
Plant disturbance
I
E
1 Z-'
/
I
I I
I I I I
I I I I 1-
Figure 8.5
A plant modeling and disturbance-canceling system that works.
To demonstrate that the system of Fig. 8.5 does work, that is, that it cau adapt and converge on P ( z ) without bias and that z-' Q ( z ) is properly obtaine for purposes of disturbance canceling, a simulation of the system of Fig. 8.5 wa comparison, a simulation of the system of Fig. 8.4 was also done. For both simulations, the plant was chosen to be
data, ensuring that steady-state conditions had been reached. Initially, the value of the adaptation constant was set at p = 0.003 for bo The results of the experiments are shown in Table 8.1. The Wienz solution, t of P(z), were compared with snapshot values of the weights of Pk(z) for bot Clearly, the system of Fig. 8.5 provided a good estimate of the Wiener solutio tion error, the average of the sum of the squares of the errors in all of the we 0,004). However, as expected, the system of Fig. 8.4 developed quite a large error (0.033). The bias is evident from the numbers displayed in Table 8.1. The adaptation constant was then reduced threefold to p = 0.001 for bot The same experiment was repeated. The results are given in Table 8.2. For the system of Fig. 8.5, noise in the weights of the plant model was red a modeling error of 0.004 to the very low level of 0.0013. This reduction is cons the expectation that for adaptive LMS processes, noise variance in the weights proportional to the adaptation constant. However, for the system of Fig. 8.4, the significant changes in plant modeling error. In fact, the error grew from 0.033 to reason, of course, is that the plant estimation bias does not depend on the speed tion. Hence, in this case, changes in p had little eff2t on the modeling error. The system of Fig. 8.5 is a very good one. If Pk(z) has the flexibility to m it will do so and the system dynamic response from point A to point C will have t function of P(z), and the plant disturbance will be oEimally canceled. It is useful to show that if, at some initial time, Pk(z) differedJy a small am P(z), then the adaptive process of Fig. 8.5 would work to bring Pk(z) toward following discussion is not a proof, but a heuristizargument. Refer to Fig. 8.6, wh plant modeling is taking place. Convergenceof Pk(z) toward P(z) for the system is assured (refer to Chapter 3 and Appendix A). It is useful to note that the transf from point A to point E, the transfer function from the input to the error point, i A
HAE(Z)= P(Z) - & ( Z ) = -APk(Z).
It is clear that malung the error smaller and smaller requires that A Pk (z) beco and closer to zero. For a given plant input X(z), the simple proportionality be z-transform of the error signal E ( z ) and -APk(z), the plant modeling error, c to the mean square error being a quadratic function_ofthe components of the v Steepest descent makes the error smaller and makes Pk (z) relax toward P(z). Th mean square error function was discussed in Chapter 3. The question is, does this also occur with the system of Fig. 8.5? Referr figure, we obtain the transfer function from the input at point A to the error at po
1
0.085 0.167 0.172
0.01 0.083 0.074
-0.013 0.042 0.05
-0.017 0.021 0.007
-0.037 0.01 -0.007
-0.037 0.005 -0.039
-0.026 0.003 -0.033
-0.023 0.001 -0.009
-0.005
-0.043 0.001
COMPARISON OF WEIGHTS OF P (z) WITH THOSE OF P^(z) FOR THE SYSTEMS OF FIGS. 8.4 AND 8.5, /L = 0.003 0.906 I 1.005
0.082 0.167 0.164
0.003 0.083 0.082 -0.040
0.042 0.039
-0.049 0.021 0.009
-0.052 0.01 -0.003
-0.065 0.005 -0.004
-0.057 0.003 0.004
-0.058 0.001 0.042
-0.051 0.001 0.012
COMPARISON OF WEIGHTS OF P (z) WITH THOSE OF P^(z) FOR THE SYSTEMS OF FIGS. 8.4 AND 8.5, p = 0.001
z) for syst. of Fig. 8.4 z) (Wiener Solution) z ) for syst. of Fig. 8.5
2
0.915 I 0.999
1 1.01 1
0.166
0.167 0.182
0.083 0.107 0.083
0.042 0.07 0.041
0.021 0.038 0.018
0.01 0.027 0.007
0.005 0.014 0.006
0.003 0.022 0.003
0.001 0.007 0
0.001 -0.003 -0.001
COMPARISON OF WEIGHTS OF P ( z ) WITH SNAPSHOTS OF THOSE OF P^(z) FOR THE SYSTEMS OF FIG. 8.7
z) for syst. of Fig. 8.4 z) (Wiener Solution) z ) for syst. of Fig. 8.5
8.3
olutionnse of P_(z), p = 0.003 nse of P ( z ) , /L = 0.001
Figure 8.6
Plant modeling.
In this case, it is clear that the z-transformof the error is not simply proportional to There are complicating factors. When APk(z) is small, the above approxima well. Assuming this to be the case, the error transform is proportional to -APk (1 - P (z) . Qk(z) . z-'). When z-I Qa(z) is very close to being a true inverse o plant noise will be almost perfectly canceled and the factor (1 - P ( z ) . Qk (z) .z very small. This-works like a low-gain factor in the error feedback loop. The re convergence of f k (z). The proper choice of the constant p is difficult to determ The system of Fig. 8.5 does work well, however. It doesjot require exte as long as the input signal is adequate for the task of adapting fk(z), and this is tage. It has the disadvantage of being more difficult than usual to work with_,bec uncertainty in the relations between p and the adaptive time constants of 9 (z) certainty in one's knowledge of the stable range of p. A suitable value of p co be found empirically however. The system of Fig. 8.7 uses external dither, and as a result, these uncer for the most part alleviated. The plant input signal needs no longer to be reli provide the proper stimulus for the plant modeling process. The disadvantage of dither is that an additional disturbance is introduced into the system. The eff disErbance can be reduced, however, if one is willing to reduce the speed of co of Pk(z). In any event, the behavior of the system of Fig. 8.7 is solid and predicta be demonstrated in the next section. The system of Fig. 8.7 is like the one of Fig. 8.3, obtaining the required ( the dither modeling scheme C introduced in Fig. 4.3(c). Although the system might seem complicated, it is easily implemented with modem day digital ha software. An analysis of this system is our next objective.
6
ak
White noise dither
D
-
%z)
r' C_OPY pk ( Z ) 1
f
\
Uk
C
# #
*
When examining the system of Fig. 8.7, one should first observe that if &(z matches P(z), the transfer function from the input at point A to the output at simply P (z), that of the plant itself. Therefore, when this system is working plant dynamics are unaltere2. Next we show that if Pk(z) departs from the value P ( z ) , the adaptive proce to pull F(z) back to P(z). The approach used above will be used again here. W finding the transfer function from point D, where the dither input to g e adapt injected, to point E, the error point in the adaptive process for finding Pk(z): ( P ( z )- &(z). A P ~ ( z ) Q. ~ ( z .z-'I ) 1 - APk(z). Q ~ ( z .z-' ) - A Pk (z) I - APk(z) . Q ~ ( z.)2-I -APk(z).
HDE(Z) =
-K ( ~ )
When A Pk ( z ) is small, Eq. (8.14) holds and the z-transform of the error of the ad cess is proportional to -APk(z), just as it was for the simple modeling scheme Accordingly, the mean square error will be a quadratic function of the compon vector APk for small APk. Steepest descent will make the error smaller by ma relax toward P (z). From this heuristic argument we conclude that for small APk(z), the dyna ior of the adaptive process in Fig. 8.7 is essentially identical to that in Fig. 8.6. F value of F , both systems will have the same weight noise statistics and the same stants. Accordingly, the pth time constant will be 1 r -- 2&'
K(z)
In order that be a good broadband model of the plant, a white dither shou As a result, all eigenvalues will be equal and the single time constant will be giv
where E [ S i ] is the dither power. When close to convergence, the weight noise covariance of &(z) will be weight noise covariance matrix
1
= COV [APk] = BI = p E [ n ; ] I ,
issue.
Referring to Fig. 8.7, the transfer function from input point A to plant o C is
Of importance is the denominator of this transfer function. This denominator acteristic polynomial of the system, and it determines stability of the disturbanc loop. Difficulty is introduced from the randomness of this polynomial becaus APk has random components. The components of APk are mutually uncorrel dividually they are correlated over time and they will vary slowly if p, as usu small. We will assume that APk(z) is quasi-consrunr over time, and that the transfer function can be used. The characteristic polynomial will therefore be u bility considerations. The characteristic equation is 1 - APk(z) . & ( z > . Z - ' = 0.
Assuming that g(z)has been constructed with n weights, n being large has essentially n components. Since these components are mutually uncorrelate are a lot of them, the transform A Pk (z) will be approximately flat for real freq the unit circle in the z-plane) and will be approximately constant over time a components of APk will all be time variable. The value of the transform of AP unit circle can be obtained as follows. The Fourier transform of the vector APk as [1 e-JW e-""
A
. . . e-("-')'"] APk = A P ~ ( Z ) J=~APk(eJ"). =~~~ +
Next consider the following expectation, in light of Eq. (8.17).
E[IAPk(eJ">1*]= E[[1 e - I W ... e-'"-"'W].APk.AP~[leJw ... e(n = [ 1 e-JW . . . e-("-"J"][covAPkl[ 1 eJw . . . e("-')'" 1 = B[1 e - J W ,,. e - ( n - l ) J w ] [ l e J W ... e ( n - l ) J w 1 7 = np
'
=np~[n:].
We have indicated above that 1 APk(el")I is essentially constant over frequenc Accordingly, it can be approximated by its expected value. Therefore
Jns.
E[lAPk(e'W)12]= nB
E [ I A P ~ ( ~ J=~ ) I ]
I
- ,i$(U)
,
'
Qk(e'")
. e - i w = 0.
Under worst case conditions, the phase factor ej'#'(w) e - j w will manage at rand fectly compensate Qk ( e j " ) at the frequency w that maximizes the magnitude o to cause satisfaction of (8.23). Let the maximum magnitude of Qk(e'"> be des Ielmax. The condition enabling the first characteristic-equationzero to reach the by increasing p (which causes /Ito increase) is therefore
Substituting (8.17) into (8.24) gives
Our conclusion is that the disturbance canceling loop could become unstable if larger than specified by (8.25). To keep this feedback loop stable, p should be cho the following range:
It is possible to kzep within this range in practice. One knows the valu number of weights of Pk(Z), and one would generally know the value of Ern: ance of the original plant output disturbance (before canceling), but getting a p estimate of 1 QIkaxtakes a special effort. Q k ( z ) is one way or another related to it can be obtained by the offline process of-Fig. 8.7. An approximate knowled would provide something to use in place of Pk(z) in an offline process for genera A nominal value for Q k ( z ) can be obtained this way. An estimate of lQlmax tained from the nominal Qk ( z ) by finding the maximum value for z = e'", sca frequency w from zero to half the sampling frequency. Accordingly,
After estimating lQliax from approximate knowledge of the plant, one can then within range (8.26) to keep the disturbance canceling loop stable. A suitable could also be found empirically. The stability of the plant modeling algorithm, the adaptive algorithm E ( z ) in the system of Fig. 8.7, is the next issue of concern. The stability issue h
P(e'") is generally flat or approximately flat at low-frequencies,the power of th Qk ( z ) is approximately given by
The stability criterion for the LMS adaptive algorithm used in the system of finding Pk ( 2 ) is, from modification of (4.29),accordingly
Values of p in this range can be chosen with knowledge of the dither power, a proximate knowledge of the input command power, the plant disturbance pow magnitude of the plant at low frequencies. In order for the system of Fig. 8.7 to be stable, it is n e c e ssq that both the canceling loop be stable and that the adaptive process for finding Pk (z) be stable. the disturbance canceling loop stability will limit p , sometimes the plant model will limit p. It is necessary therefore that p be chosen to satisfy both (8.26) an
8.7
ANALYSIS OF PLANT MODELING A N D DISTURBANC CANCELING SYSTEM PERFORMANCE
We have thus far determined the time constants and weight covariance noise fo have approximately determined the stable range of p for the system of Fig. 8. objective is to determine the components of signal distortion and noise that ap plant output due to dither, uncanceled plant disturbance, and the effects of adap in the weights of g ( z ) . Refemng to Fig. 8.7, the transfer function from the dither injection point D output point C is
practice. Next, the effects of the random A Pk (z) will be taken into account. The tra tion from the input u’(z) at point A to the plant output at point C is H A C ( Z= ) %
P(Z)
I - A P ~ ( z* )Q ~ ( z.z-’ )
P(Z)+APP~(Z),Q~(Z).P(Z).Z-’.
Equation (8.32) applies when APk(z) is small, which is the usual operating cond first term of the right-hand side is P ( z ) , and this is the normal transfer function fo V ( z )to encounter in propagating through to the plant output. The second term tional to APk(z), which varies slowly and randomly. This component of the tra tion causes a randomly modulated component of output signal distortion. We are with its total power. For ease of computation, we will assume that Z-’
. Q ( z ) P ( z ) * 1. 9
Therefore, the transform of the output signal distortion is approximately given b transform of
= U ’ ( z ) . APk(z). distortion This can be written in the time domain in vector form as output distortion From this we obtain the power of the output signal distortion: [ output
We assume that the fluctuations of APk are independent of the input U’, and tha tistically stationary. The covariance of APk is given by (8.17). Accordingly,
distortion
-
= E[U’T.E[APk AP:]. U’]
= E[U’T* cov [APk] U’]
(8.11). The power of the uncanceled plant output disturbance is, in accord with uncanceled plant output disturbance power
= 4cc (0).
The overall output error power for the system of Fig. 8.7 is the sum of the o power (8.31), the output signal distortion power (8.37), and the uncanceledplant turbance power (8.38). Accordingly,
Using Eqs. (8.16) and (8.17), the above equation may be expressed as
After choosing a given value of time constant r for the adaptive process for & (t all output error power can be minimized by best choice of the dither power. derivative of the right-hand side of (8.40) with respect to E[$]to zero, we obt
Therefore,
C will always be a fraction The uncanceled plant output disturbance power ~ C (0) output noise power ~ [ n : ] . For the plant modeling and noise canceling system of Fig. 8.7, Eqs. (8.41 are useful indicators of system performance. Once an adaptive time constant r Eq. (8.41) indicates how to select the dither power. The value of I.L is selected, w edge of E [ $ ] and r , by using Eq. (8.16). In any event, the determination of p
In order to verify workability of the plant modeling and disturbance canceling Fig. 8.7 and its analytic treatment, a set of computer simulation experiments formed. Of greatest importance were
(i) Verification of the convergence of & ( z ) to P ( z ) with time constant (8.16). (ii) Verification of the stability conditions of (8.26) and (8.29). (iii) Verification that output noise power and signal distortion are redu greatest extent possible. Verification of Eqs. (8.41) and (8.42). (iv) Verification that the overall transfer function remains essentially un the disturbance canceling feedback.
As a first step, however, we yanted to verify that the system of Fig. 8.7 converge to an unbiased estimate P ( z ) of the plant P ( z ) . For this purpose, w the same simulation that was performed in Section 8.4 for illustrating the beha systems of Figs. 8.4 and 8.5. In other words, we have assumed that
The plant disturbance nk was created by inputting a white noise to a 20-weigh moving average filter with ~ [ n i=] 0.2.
The plant input u; was zero-mean, white, and of unit power. However, in con simulations of Section 8.5, the plant modeling was performed by dithering sch depicted in Fig. 8.7. The dither was chosen to be white, zero-mean, and of unit power. Keepin parameters constant, we were able to vary the adaptation constant p . For each p, the system was run for 5,000 iterations to ccme to equilibrium, and then in e snapshot was taken of the impulse response of P ( z ) , with the results presented in Referring to Table 8.3, lize 1 shows the first 10 impulses of plant P ( z ) (or t solution for_the plant model P ( z ) having 10 weights). In line 2 are presente weights of P ( z ) while adapting with p = 0.003. Clearly the match between th plant model and the Wiener solution is quite good (square error was only 0.00 p = 0.001, the plant model (as depicted in line 3) was even closer to the Wien (square error was only O.ooOo237). We can see from this that the system of Fi capable of delivering an unbiased, accurate plant model.
Accordingly, for the stability study we choose a new plant, 9
For this plant, (8.26) and (8.29) gave completely different stability ranges. The described above was repeated for this plant. As before, the plant disturbance the plant input power were kept constant. The parameter p was increased until became unstable. For the first test, the plant disturbance power was set at 0.2. Fo test, the plant disturbance power was raised and set at 20. For both tests, the power was 1. The results of the stability study, both measured and theoretica in Table 8.4. The maximum theoretical value of p permittitg stability is in ea smaller of the two limits obtained from (8.29) (stability of Pk(z)), and (8.26) the disturbance canceling loop). Inspection of Table 8.4 shows close correlati the theoretical predictions and the experimental results. TABLE 8.4 STABLE RANGE OF p FOR THE SCHEME OF FIG. 8.7, FOR TWO VALUES OF PLANT DISTURBANCE POWER
Plant disturbance power
Maximal stable range of 11 from experiment
Maximal stable range of p from Eq. (8.26)
Maximal stable range of p from Eq. (8.29)
0.08
0.18 0.0018
0.050 0.045
0.2 20
0.003
The theoretical predictions were obtained in thz following way. In orde
Eq. (8.26), we needed to evaluate IQliax. Given that P ( z ) = P ( z ) (they both ha
number of weights), the Wiener solution for Q ( z )was Q(z) = 0.701
- 0 . 6 5 7 ~+ ~ 0.176~-' ' -0.132~-~.
The maximal value of I QI2 is obtained for z = - 1. Then,
I Ql =i, ( 1, .667)2= 2.78. Since the number of weights of P^(z) is n = 10, Eq. (8.26) is I (10)(2.78)E[n$
>p>o.
Hence,
Accordingly, (8.29) can be expressed as I >@LO. lO(1 1 (O.OI)E[n$)
+ +
This expression was used in order to compute maximal stable values of p for E and E[n:] = 20. The results are given in Table 8.4. Inspecting the first row of Table 8.4,we conclude that for small values of turbance power, stability condition (8.29) is the dominant one. Inspecting the s of Table 8.4, we conclude that for cases with large plant disturbance power, th of (8.26) becomes dominant. For both cases, theoretical prediction of the stable range of p matched the e tal range to within an error of 40 percent. The theory was somewhat more conserv necessary to ensure stability. Next we wanted to verify Eqs. (8.16), (8.26), (8.29), and (8.42) for a non phase plant. For these simulations, a simple one-pole plant with a unit transport chosen:
This plant is nonminimum-phase. The plant output disturbance nk was chosen pass Gaussian and was generated by applying white Gaussian noise to a single The plant input signal u; was a sampled low-frequency square wave. Its freq approximately 0.0008 of the sampling frequency. The adaptive filter Q k ( z ) w only three weights (increasing the length of Q k (z) caused only negligible impro disturbance canceling performance because of the low-frequency character of th turbance). To establish convergence of to P ( z ) , a plot of IIAPklI versus numb tations was drawn and this is shown in Fig. 8.8. This is a form of learning curve retical time constant was t = 2,500 sample periods. The measured time constan from the learning curve was approximately 2,000 sample periods. Convergence with very low residual error. The dither power was fixed at unity for a stability checkinkexperiment. T was run with p gradually increased until the adaptive process for Pk.( z ) blew up. runs was made, each with a different value of plant disturbance power. Both stabi
E(z)
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Figure 8.8
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A learning curve for adaptive plant disturbance canceler.
were checked. The theoretical stability limits for did work, but they were n confirmed, only verified within a factor of 2 or 3. The results of a plant disturbance canceling experiment are shown in Fi plant itself without disturbance was subjected to a square-wave input, and the sponse, almost a square wave, is shown in Fig. 8.9(a). The same plant with the was then set up with low-pass plant disturbance added to its output, and the resu in Fig. 8.9(b). Next, this same plant with the same low-pass plant disturbance a was connected for disturbance cancelation in accord with Fig. 8.7. The dither w and the same square wave was input as u ; . The dither power was optimized in (8.41). The plant output is shown in Fig. 8.9.c. The effects of plant disturbance are in clear evidence. The plant output disturbance power was reduced by 18 (8.42) to calculate the overall output error power, the plant output disturbance po have been reduced by 19.2 dB. Many more experiments have been made in addition to this one with equall ification of these formulas being the result. Preservation of the plant’s dynamic evidenced by the similarity of the waveforms of Fig. 8.9(a), without disturbanc and Fig. 8.9(c), with disturbance canceling in operation. One more experiment has been performed in order to demonstrate in gr how the dynamic plant response is preserved when the adaptive plant disturbanc system of Fig. 8.7 is in operation. The results are shown in Fig. 8.10. A step applied to the disturbance-free plant, and the output response was plotted. The with additive output disturbance was then tested. With the disturbance cance working and with the dither applied, the input u; was once again a unit step. Th was recorded. This was repeated many times. An ensemble average of step r plotted in Fig. 8.10. The ensemble average and the step response of the original d free plant are hardly distinguishable, indicating the preservation of the plant res original form even while the adaptive plant disturbance canceling system was o
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Figure 8.9 (a) Square wave response of undisturbed plant; (b) Square wave response of dist (c) Square wave response of disturbed plant with adaptive disturbance canceler.
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Figure 8.10
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Demonstration of the preservation of plant dynamics even when canceling plant
APPLICATION TO AIRCRAFT VIBRATIONAL CONTR
Adaptive plant disturbance canceling can be used to alleviate a problem that so flicts passengers of aircraft, the problem of sudden changes in altitude and a due to vertical wind gusts. Reduction of the variance of the airplane’s respons dom wind loading could greatly increase comfort and safety. An airplane subjec updrafts and downdrafts is sketched in Fig. 8.11. - Aileron Control Signal
‘X c
‘
Accelerometer
Aileron Servo
Aileron
iiiiliii UpDowndrafts Gusts
Figure 8.11
An airplane with gust loading canceler.
Corrections in altitude could be achieved in a conventional way by con elevator at the tail of the airplane. By causing the tail to depress, the angle of att and the airplane climbs. By raising the tail, the airplane descends. Correcting f
.
in differential mode, that is, when one aileron goes up, the other goes down. P aileron down increases the lift on its respective wing; at the same time, the ot goes up and reduces the lift on its wing. The torque causes the airplane to roll, a banking turn. We are proposing here that the linkages and servos which actuate the ailero ified so that the ailerons could be driven in a differential mode for steering and time in a common mode for rapid variation of wing lift. The differential and com signals would be summed at the inputs of the two servos driving the ailerons. Th tial mode signals would be the usual steering signals, while the common mode sig be coming from an adaptive disturbance canceler rigged to suppress the effect updraft and downdraft wing loadings. The random forces acting on the airplane can be treated as plant disturban acceleration can be sensed with a suitably placed accelerometer. The ailerons c ulated to apply compensating wing forces to reduce vertical acceleration. For th the accelerometer output can be regarded as the plant output. The common mo the aileron servos can be regarded as the plant input. The plant itself is a dyna comprised of the ailerons, the wings, the airframe, and the surrounding air flu for only one of the aileron servos is illustrated in Fig. 8.1 1 . Control for the oth identical. Figure 8.12 shows the adaptive control system for minimizing the variance tical accelerometer output. This is a more succinct form of the same adaptive p bance canceler that is shown in Fig. 8.7. The dithering method is based on schem particular configuration shown in Fig. 4.3(d)). Q ( z )could be obtained offline a Fig. 8.7. The dynamic response of the airplane to aileron control signals rema the same, with or without the adaptive canceler. The basic handling of the airplan stays the same. The acceleration canceler simply reduces the bumpiness of the flying through choppy air. A multi-input multi-output (MIMO) version could b using appropriate sensors and actuators, to cancel random disturbances in yaw roll, MIMO systems are discussed in Chapter 10. Control of vertical disturbance in the presence of vertical wind gusts has n done with adaptive inverse control. The U.S. B1 bomber does have vertical d control based on a more conventional servo technology applied to a pair of sm wings on the forward part of the aircraft. Without this control, the ride would b under certain flight conditions that the crew would not be able to physically func disturbance control needs to be developed for civilian aircraft now. Let’s do it!
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& Dither
Error
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U’(Z) Aileron control signal from autopilot I
I I I
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4 COPY
I-’
Adaptive disturbance canceler
------,-,-,,,,-,-,,,-----J
Figure 8.12
8.10
An adaptive canceler for minimizing gust loading effects.
APPLICATION T O EARPHONE NOISE SUPPRESSION
In Section 7.6 we discussed the problem of earphone noise cancelation. Anoth to the problem is illustrated in Fig. 8.13. This approach uses only one micropho located between the earphone and the ear cavity. This is microphone M2. The m disclosed here is based on the plant noise canceling idea embodied in the system A block diagram of the entire new system is shown in Fig. 8.14. To understand how this system works, it is useful to trace through it o a time. The microphone signal labeled M2 is the sum of two components, “p bance” (due to (SAN), the strong ambient noise through the transfer function [ and “plant dynamic response” (the microphone output in response to the earph The “plant” here is equivalent to the transfer function [M2IEPS].The earphone s audio input signal (AIS) minus the filtered plant disturbance, derived from the no nent of the microphone M2 signal. The adaptive process for finding the plant m offline process for finding Q, and the disturbance canceler work just like the cor processes of the system of Fig. 8.5. The earphone signal (EPS) connects to th and the microphone signal M2 connects into the adaptive canceler as shown in The audio input signal (AIS) connects to the canceler as shown in Fig. 8.14.
Figure 8.13
An alternative method for cancelation of earphone noise.
The system of Fig. 8.14 is the best canceler for the earphone and microph uration of Fig. 8.13. No linear canceler can do better. The question is whether this is a better system than the one presented in The situation is not clear, but the answer is probably no. The system of Chapte microphones and works on a different principle. It is probably a better system uses two microphone sensors instead of one, (i.e., there is more information to disturbance canceling) and because the outer microphone M I receives the stro noise earlier than M2,making it easier to cancel the ambient noise components through the earphone. There is no doubt that both the noise canceling schemes of Chapters 7 and well. Systems like them have been analyzed and simulated without difficulty. T ing characteristics can best be obtained by building the two systems in hardwar paring them critically under real-life operational conditions. If these systems co cheaply enough, they could be used on commercial aircraft where they would greatly to the enjoyment of in-flight movies and music.
CANCELING P L A N T DISTURBANCE FOR A STABILIZ MINIMUM-PHASE PLANT
The stabilized minimum-phaseplant that was studied in Chapter 7 was simulated to illustrate plant disturbance canceling for this case. Figure 8.15 shows a block the experiment. It is basically the same block diagram as the one in Fig. 8.5, exc customized for this plant of interest. It also includes a perfect G(z), the discreti of the true stabilized plant transfer f ~ n c t i o n The . ~ output of this G ( z ) is the id the discrete output of the plant if there were no disturbance and no disturbanc system. Comparing the plant output with the ideal output yields the residual e
'In a real-world situation, G ( z )would usually not be perfectly known. G ( z )is assumed to be kn test purposes only.
Audio input signal
T
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(A19
Earphone signal (EPS)
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Noise
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. . . OfRine . . . process . . . for. generating . . . . .Q (.z ). . . . . . . . . . .
Figure 8.14
Block diagram of alternative method for cancelation of earphone noise
celing began then. The input driving function U ( z ) was a zero-mean first-ord process, and the plant disturbance was an independent zero-mean first-order M cess. The measured power of the residual error was reduced by a factor of 5.23 i This result was very close to the theoretical optimum. Figure 8.19 shows the pl and the plant output when near optimally canceling the plant disturbance. When G ( z )is unknown, adaptive identification is used to obtain &z), as il Fig. 8.15. This was done by means of DCTLMS. At one moment in time, the ad cess was halted and the impulse response of 6 ( z )was obtained, and it is plotted in Corresponding to this impulse response, the offline process was used to obtain Q impulse response is shown in Fig. 8.21. These impulse responses are basically sions of the ideal ones shown in Figs. 8.16 and 8.17. A plot of the square of th error is shown in Fig. 8.22. The canceling feedback loop was engaged at the 5, sample, and the power of the residual error dropped by a factor of 5.03. This is n as optimal, but it is close. A comparison of the plant output with the adaptive di canceled output is shown in Fig. 8.23. Using a real-time adaptive disturbance ca results are not so different from those that would be obtained with a near optim having perfect knowledge of G ( z ) and its near optimal Q ( z ) . A series of tests was performed next with other kinds of plant disturbanc step and ramp functions. For one of these tests, driving function U ( z )was Markov There was no Markov disturbance. At time sample 2,000, the canceling feedback on and at time sample 3,000, a step disturbance was turned on. This simulates a with a sudden onset. The step disturbanceis plotted in Fig. 8.24. The residual erro in Fig. 8.25. There was no residual error until the canceler was engaged. Then th small. At time 3,000, the error had a sharp spike because the canceler could no instantaneous disturbance, the leading edge of the step. But within one sampl residual error was brought down to low level and it remained at low level with a close to zero. Figure 8.26 shows a comparison of the plant output with the ideal p in a steady-state situation, after the leading edge of the step disturbance has pas the time window of the plot. In many respects, the essentially zero steady-state e presence of a constant disturbance represents behavior like a type 1 servo. Other tests were done with ramp disturbances, and the results were quite ramp disturbance was turned on at time sample 3,000, and allowed to plateau at ple 4,000 as shown in Fig. 8.27. The disturbance canceling loop was closed at ple 2,000, and a small residual error is evident in Fig. 8.28 from then on. Whe disturbance commenced at time sample 3,000, the effect on the residual error w imperceptible. The same was true at time sample 4,000 when the ramp saturate parison of the actual plant output and the ideal plant output in the middle of the ra
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Impulse response of G ( z ) ,the discretized transfer function of the stabilized minimum
Figure 8.17
Impulse response of the near optimal Q(z) for C ( z ) .
Figure 8.18 Square of the residual error plotted over time. Canceling feedback loop closed at tim ber 5,000. Q ( z )was obtained from G(z).
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Figure 8.19 A comparison of the plant output and the output of the minimum-phase plant wit disturbance canceling.
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Figure 8.20
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Time (0.1 sec.) Impulse response of &I) for stabilized minimum-phase plant.
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Impulse response of Q ( z ) for e ( z ) .
Figure 8.22 Square of the residual error. Disturbance canceling began at the 5,000th sample ti obtained from G ( z ) .
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Figure 8.23 cance1er.
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Plant output and plant output for the minimum-phase plant when using an adaptive pla
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A step plant disturbance.
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is shown in Fig. 8.29. A similar comparison is shown in Fig. 8.30 in steady stat the ramp has plateaued. There is not much difference between the actual and idea with a ramp disturbance. This behavior is like that of a type 2 servo.
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Figure 8.28 Residual error of minimum-phase system. The canceling loop was turned on at time The ramp commenced at time sample 3,000 and plateaued at time sample 4,000.
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Flpre 8.30 Comparison of actual and ideal plant outputs for minimum-phase system with ram Data were taken after the ramp saturated.
8.12
COMMENTS REGARDING T H E OFFLINE PROCESS FOR FINDING Q(z)
The offline process could make use of an adaptive algorithm to find Q(z), o Wiener theory to find a solution. The adaptive process could be based on LMS, LMS,or some other algorithm. Experience has shown, however, that a more co ally efficient approach to use offline when utilizing a computer to find Q ( z ) is approach. If adaptive hardware were available on the other hand, then an ada rithm would be appropriate. For the above example, Q(z) was calculated by co computer, and the Wiener approach was used. Refemng to the block diagram of Fig. 8.15, the offline process is drive whose power spectrum should be similar to that of the actual plant disturbance. of power spectrum is not critical. In practice, one would generally not know the spectrum exactly. One would use a best guess. For thz above example, we kne trum of the disturbance. Since in addition we knew G ( z ) ,it was possible to c spectrum at the input to the filter Q ( z ) by making use of Eq.(2.22). Transformi autocorrelation function. From the autocorrelation function, it was possible to c R-matrix for the inputs to the weights of the filter Q(z).The P-vector was com the crosscorrelation function between the weight inputs and the desired respon thaGs, the driving noise. This was done by making use of (2.15) and (2.14) with of G ( z ) .The Wiener-Hopf equation that we used was obtained from (3.1 1) as RW* = P.
Having R and P, Gaussian elimination was used to solve the linear equations an thus yielding the weights of Q ( z ) .No matrix inversion was involved.
disturbance was step and ramp. The amplitude of the bias is not critical, but sho be comparable in amplitude to the standard deviation of the noise. Using both bias to drive the offline process, the resulting canceling system works well whe disturbance is random with a finite mean. When comparing the block diagrams of Figs. 8.15 and 8.3, the offline p canceling loop of Fig. 8.3 contain a unit delay between the plant model and Q offline process and canceling loop of Fig. 8.15 contain no corresponding delay. T only necessary when the plant impulse response has a finite component at zero ti responds instantly. For the minimum-phaseexample, the plant could not respon because it had more poles than zeros in the s-plane. The stabilizing feedback do this. The stabilized plant, when discretized, had a unit delay in its impulse res additional delay in the offline process was not necessary and its inclusion would h a loss in performance for the disturbance canceler. Therefore, the offline proce disturbance canceling loop of Fig. 8.15 did not incorporate an additional unit de
CANCELING P L A N T DISTURBANCE FOR A STABILIZ NONMINIMUM-PHASE PLANT
The stabilized nonminimum-phaseplant that was studied in Chapter 7 is once ag ject of study. We will demonstrate disturbance canceling for this system. A bloc of the adaptive canceler is shown in Fig. 8.31. The exact impulse response of lized plant is shown in Fig. 8.32 with k = 24. Its transform is G ( z ) . For this offline process is used to obtain Q ( z ) . Its impulse response is shown in Fig. 8 used in the canceling system, a reduction in the residual error takes place. The ex reduction depends on the characteristics of G ( z ) , Q ( z ) ,and on the spectrum o disturbance. With a nonminimum-phaseG ( z ) ,low-frequencydisturbances are m to cancel than high-frequency ones. Results of a disturbance canceling experiment are shown in Fig. 8.34 when ideal G ( z ) and Q(z). The theoretical reduction in residual error power is 5.65. results came very close to this. Figure 8.34 shows a time plot of the square of th error. The canceling loop was closed at time sample number 5,000. Figure 8.35 actual plant output and the ideal plant output after closure of the plant disturbance loop. The difference between these plots is the residual error. An adaptive canceler was tested next. The DCTLMS algorithm was used c ( z ) . An instantmeous impulse response is shown in Fig. 8.36. This is a slig version of the ideal impulse response shown in Fig. 8.32. The corresponding i sponse of Q(z), obtained by the offline process block-diagrammed in Fig. 8.31
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Figure 8.33 Impulse response of Q ( z ) , optimized for G ( z ) ,the exact discretized transfer function lized nonminimum-phase plant with k = 24.
F b r e 8.34 A time plot of the square of the residual error. The canceling loop was closed at sam G(z) and Q(z) were used. Nonminimum-phase plant, stabilized with k = 24.
Time (0.1 sec.)
Figwe 8.35 Actual plant output and ideal plant output after closure of disturbance-canceling lo and Q ( z ) were used. Nonminimum-phase.stabilized with k = 24.
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Figure 8.36 An impulse response corresponding to c ( z ) obtained by DCTiLMS. Nonminimum stabilized with k = 24.
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Figure 8.37 Impulse response of Q(z) corresponding to G ( z ) obtained by DCTLMS. Nonmi plant, stabilized with k = 24.
Figure 8.38 The square of the residual error. The loop was closed at sample 5,000. Adaptive c were used. Nonminimum-phasestabilized with k = 24.
c(z)
This experiment convinces one of the importance of having be as c sible to G ( z ) .$low adaptation is the way to achieve this, and one should make of weights of G ( z )be no greater than needed to make a good fit.
INSENSITIVITY OF PERFORMANCE OF ADAPTIVE DISTURBANCE CANCELER T O DESIGN OF FEEDBAC STAB ILIZAT I0N
In Sections 8.11 and 8.13 we analyzed two cases of plant disturbance canceling plants. However, since inverse control and disturbance canceling cannot be appl to an unstable plant, conventionalfeedback stabilizationhas been utilized first. T is: How has the overall system performance been affected by the particular ch stabilization feedback? In other words, is it possible to have further reduction output disturbance power if different stabilization feedback is chosen? This question is answered in Section D.2. It is shown there that, as long a lization feedback itself is stable, the plant output disturbance power is indepen
Figure 8.39 Ideal plant output and actual plant output with canceling loop closed. Adaptive used. Nonminimum-phase plant was stabilized with gain set at k = 24.
c(z)a
choice of stabilization feedback. This result is very fortunate. It means that usin tive disturbance canceler for control of unstable plants does not create difficult p optimal feedback design.
SUMMARY
This chapter discussed methods for optimally reducing plant disturbance withou plant dynamics. Feedback is used to bring plant output disturbance back to the p in order to effect plant disturbance canceling. The best linear least squares plant d canceler may be configured this way. Although the disturbance canceler is a fee tem, the transfer function around the loop is essentially zero. For this reason, th response of the plant with or without disturbance canceling feedback is the same When using the system of Fig. 8.7 with white dither, the learning process fo only one time constant, given by
where E [ $ ] is the dither power. When using an offline process for finding &(z speed of convergence of the plant modeling algorithm and of the plant disturbanc are the same and this speed is determined by (8.16). The disturbance canceling feedback has effect on both the stability of th modeling process for finding P ( z ) and of the disturbance canceling loop itself. F
frequency. Stability of the plant modeling process of Fig. 8.7 is assured by keeping j range 1
+
(
n[E[8,21+ E[(4)21 mi1
>P>Oo,
,p(;,"),f)w
,owl
where E [ ( u ; ) ~is] the power of the plant input signal and I P ( e j w ) ( i,ow is the squ nitude of the plant transfer function at low real frequencies. By %ems of a heuristic proof, it was demonstrated that the equilibrium the vector P is the Wiener solution P.The equilibrium position and the Wiener not affected by the presence of the disturbance canceling feedback loop. Concluding the analysis of the system of Fig. 8.7, the overall output erro defined as the sum of the dither power at the plant output, the output signal disto caused by the modulation of the feedback loop signal due to gradient noise in th P,and the uncanceled plant output disturbance power. The optimal value of d is the value that minimizes the overall output error power. This is given by
A
where E [ ( u ; ) ~is] the power of u i , the input to the plant and its disturbance in Fig. 8.7, and xyd p' is the sum of squares of the impulses of the impulse P. When using the optimal dither of (8.41), the minimum value of the overall power is
, [;g:)/ = power
2nE[(u')2]. E [ n 3
cz p'
+ 4cc (0) t
mi"
where C#JCC(O)is the power of the uncanceled plant disturbance, defined by (8.1 These formulas were challenged with a number of simulated experimen found to work well for all of the cases tested. Adaptive canceling was applied to stabilized minimum-phase and no phase plants. The depth of disturbance canceling was generally much greate imum-phase plants. In any event, the linear adaptive disturbance canceling scribed in this chapter was optimal in the least squares sense. i t was shown th ity to cancel disturbance was not affected by the design of the plant stabilizer.
System Integrati
9.0
INTRODUCTION
Many different control systems can be configured from the parts that have been dev in the previous chapters. Each system would have an adaptive inverse controller con to a plant to be controlled. The plant may or may not have an attached adaptive distu canceler. Among these systems, one has been selected for discussion here and it diagram is shown in Fig. 9.1. In this system, the plant uses an adaptive disturbance c like that of Fig. 8.7. The controller and the offline Eethod for obtaining it are the s those of Fig. 6.4.The adaptive process for finding Pk(z) incorporates dithering sch This system was built in software and tested extensively. Although it has man it turns out to be a solid, robust system. A straightforward start-up procedure is lowing: Use the panic button to break the disturbance canceling feedback loop unt converges to P ( z ) , then close this loop. In the event of a sudden catastrophic ch P ( z ) that would cause the entire system to go unstable, push the panic button unt converges to the new P ( z ) . Without the disturbance canceler, stability is much le consideration. Basically, the system will be stable as long as the plant P ( z ) is stabl Several questions remain about this system: the choice of dither power, the ch v, and the resulting time constant of the adaptive process. Finally, the output error needs to be evaluated.
9.1 OUTPUT ERROR AND SPEED OF CONVERGENCE
There are three sources of output error or output noise: (a) uncanceled plant distu (b) dither noise, and (c) effects of noisy A Pk(z) and ACk(z) modulating the comma nal I (z) while it is propagating through the entire system. These errors have alread analyzed for the various parts of the system. The goal now is to put the pieces tog determine overall system performance. We begin with a consideration of item (c) above. There are some surprises ab effects of A Pk (z) and ACk (z) on the random modulationof the input I ( z ) while prop through the system. The transfer function of the overall system from input to outpu
258
* COPY ck(Z)
I Controller
4
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Plant * output
-
(2
1P ( 2 )
P ( z ) + A P k ( z ) - APk(Z). P(z).z-'Q~(z)-[AP~(z)
The effects of both A P k ( z ) and ACk(z) are incorporated in this expression A Pk( z ) , we can neglect the last term in the denominator.
Assuming that
then
From Eq. (9.6), one concludes that a change in A P k ( z ) causes only a second-o in H,o(z) and in turn, only a second-order effect on the plant output signal. Ca the first-order effects of APk(z) and ACk(z) takes place as a byproduct of the plant disturbance canceling system. This conclusion, from Eq. (9.6), is based o which is easily met in practice, and on l3q. (9.5) which can be closely met with phase plant and approximately met with a nonminimum-phaseplant. The output noise is therefore the sum of the uncanceled plant disturba dither noise. The power of the uncanceled plant disturbance is &--(O), in acco (8.10) and (8.38). The various factors that enter into this residue of the plant have been described in Chapter 8. The dither noise at the plant output was deriv ter 8, for scheme C as (8.31): plant
power Therefore, the total plant output noise is
i =O
be possible because of stability considerations, but making E[Sz]large causes lar the plant output. One would generally like to be able to adapt rapidly and at the have low plant output noise power. Stability is the most important consideration Stability criteria have been obtained as (8.26) and (8.29) for the system o The same criteria apply to the system of Fig. 9.1. The presence of ? k ( z ) as the in Fig. 9.1 has an effect on stability in light of criterion (8.29), since the power o input Uk is important to this criterion. The various factors need to be properly to use (8.26) and (8.29) to calculate the stable range of p. Once the maximum estimated, a practical value of p can be selected by cutting down from this max suitable factor of safety. With p selected, the dither power may then be selected. It is clear from (8 small dither power will result in a large time constant, and that by increasing dit the adaptive process will converge more rapidly. Reducing the dither power will output dither noise but slow the adaptation process. Since the uncanceled plant d is irreducible, there is no point in making the output dither noise power substa than the uncanceled plant disturbance power. A reasonable compromise betwe jectives of fast adaptation and low plant output noise would be to make the dithe the plant output equal to the uncanceled plant disturbance power, that is, to ma terms of (9.8) equal. As such,
Now having selected p and the dither power, the time constant of the adaptive given by (8.16). The effects of dither power on system performance have already been test mentally for the system of Fig. 8.7. The addition of the controller in Fig. 9.1 w only negligible effect on these results and the experiments will not be repeated same is true of the previous stability studies.
SIMULATION OF A N ADAPTIVE INVERSE CONTROL SYSTEM
A number of simulated experiments were performed to test the behavior of the Fig. 9.1 and to compare its behavior with theoretical predictions. The plant chose experiments was FIR and minimum-phase, and it had the transfer function of P(z) = (12-1) i .
2 i=O
Utilizing the system of Fig. 9.1, we were able to achieve, simultaneously, of the plant disturbance and excellent control of plant dynamics. At the beginzing, we wanted to simulate the effects of imperfect plant mo cordingly, we set P ( z ) to be ~ ( ~ ) + o . o i ( i + ~ - ~ + . . ,. + ~ - ~ )
and ceased plant model adaptation. The dither was shut off, and the plant distu shut off. Clearly, as a result of this deliberate bias in the plant model, an erroneou was computed. The error in the Controller, in turn, might cause some excess plant output. In order to validate our prediction that the plantjisturbanze canceler wi eliminate distortion caused by error in the weights of P ( z ) and C ( z ) ,we ran th Fig. 9.1 sometimes with the disturbance canceling loop open, that is, Q k ( z ) = 0 for all k,
and at other times with the disturbance canceling loop closed. We have consi cases:
a. The plant P ( z ) was controlled by a fixed 10-weight Wiener controller ba fect knowledge of the plant P ( z ) . The plant disturbance canceler was dis
b. The plant P ( z ) was co_ntrolled by a fixed 10-weightWiener controller b biased plant estimate P given by (9.13). The plant disturbance canceler w nected.
c. The plant P ( z ) was controlled by the same fixed 10-weightWiener contr used in (b), based on the biased plant estimate. Although there was no p bance, the plant disturbance canceler was connected. Q ( z ) was fixed, b same biased plant estimate.
In all three cases, the overall system impulse response (the convolution o its controller) was found and compared with the impulse response of the refer M ( z ) . The resulting impulse responses (truncated to their first 11 impulses) ar in Table 9.1. Also, the output distortion power (i.e., the output error power that both plant disturbance power and dither power are equal to zero) was measured cases. The first line of Table 9.1 gives the impulse response of the reference mode with (9.2). The second line gives the overall impulse response obtained in ca for the best 10-weightcontroller based on perfect knowledge of P ( z ) ) . The ma
1
0 0
2
-0.005
0
0
3
0
-0.005
0 0
0
-0.005
0
0
-0.005
0 0
0
-0.005
0 0
Impulse Number 6 7
0
-0.005
0 0
8
0
-0.005
0
0
9
0
-0.005
0 0
10
0.00 1
-0.003
0 -0.00097
I1
5
1 1
-0.005
0
4
0.99 0.005
0
0.99
PULSE RESPONSES OF ADAPTIVE INVERSE CONTROL SYSTEMS WITH BIASED PLANT MODEL, WITH AND WITHOUT BANCE CANCELING
se of Reference Model se for Ideal Controller se for Biased Plant Model, e Canceling se for Biased Plant Model, nce Canceling
(
variance of output error for controller based on biased F(z)
1
= 2.82(
Line 3 of Table 9.1 shows the overall impulse response for case (b), and it d strongly from the reference model impulse response than does the impulse r case (a). Finally, the disturbance canceling loop was closed. Line 4 of Table 9.1 overall impulse response for case (c). By comparing lines 3 and 4, we can see troduction of plant noise canceling indeed reduced the output error. The new er was variance of output error for controller based on biased p(z), = 1.42( l o r 4 . with plant disturbance canceler on
1
Comparing (9.17) with (9.16) confirms the fact that the effects of the e controller, due to plant model errors, tend to be canceled and mitigated by co errors in the plant disturbance canceling loop. The variance of the output error w by more than an order of magnitude by the disturbance canceling loop. Note that error cancelation was not perfect, however. The output error case (c), given by (9.17), was higher than that for case (a), given by (9.15). In controller was almost ideal. There are three reasons that limit the capability o bance canceler to cause the effects of A P and A C to cancel. These are
1. A P is supposed to be small. If it is not so small, second-order effects co cause output distortion;
2. The feedback filter Q has only a finite length, chosen by the system desi
3. The disturbance canceling feedback includes, by its nature, one unit of result, the disturbance canceling loop cannot correct any error in the firs the impulse response.
In order to improve the diprbance canceler and enhance the cancelation system error, one could adapt P with smaller p to reduce weight noise, one longer Q where this is indicated to be advantageous, and one could run with a pling rate so that error in the first samples of the impulse response would be less Doing all this would make the variance of the dynamic system error extremely
system error, is based on the concept of transfer function. For the transfer functio or of a system to be algebraically meaningful, %is necessary for that system to be time invariant. While adapting, it is c l y r that! is not time invariant. Although tion of Eq. (9.6) for the time variable P and C is not strictly valid, if one were with a small w , this would cause AP and AC to change slowly and would make transfer function be reasonable. In any event, experimental evidence seems to in the effects of AP and AC upon the dynamic system error cancel while adapting or at even moderate speeds. With the basic system of Fig. 9.1 operating and adapting normally, experim done in the following manner. In addition to the basic system, several forms of aux tems were sizulated havmg the same architecture as the basic system but with Pk, Q k , and C not adapting normally. Their weights were obtained by copying tive weights from the basic system. Each auxiliary system was used to explore critical output components of the basic system. For example, one of the auxilia was used to determine and measure the plant disturbance at the control system ou was done by turning on the plant disturbance in the auxiliary system but leaving and the command input signal off. The output of the auxiliary system containe ponent of interest. Another auxiliary system was used to observe and measure component of the control system output. This was done by turning off the comm signal and the plant disturbance. and turning on the dither. Another auxiliary s used to explore the dynamic response of the control system to the command inp thing was turned off except the command input, and the output of the auxiliary s the dynamic response being sought. This dynamic response was able to be com checked against the output of the reference model. The distortion components in this dynamic response which are due to AP a supposed to cancel from the action of the plant disturbance canceler. An addition this idea was accomplished by comparing the dynamic responses at the output iliary system when the disturbance canceling loop in the auxiliary system was and when it was disconnected. We have done all of these experiments in order to check the ability of the d mulas to predict distortion in the control system output due to uncanceled plant d dither noise, and dynamic response distortion. Running the basic system of Fig. 9.1 in a normal way and running an auxili at the same time with the dither and command input off, the power of the uncan disturbance was measured to be
A
0.0395.
With the command input and plant disturbance turned off in the auxiliary the white dither with unit power turned on, the power of the dither in the outpu iliary system was measured to be 1.2818.
A formula for the dither noise in the plant output is &. (9.7). Since the param equation were known perfectly, the dither output power was calculated theoret 1.3333.
To study dynamic response distortion, the auxiliary system was operate disturbance and dither turned off, while the command input was turned on. system, the value of 1 was chosen to be O.OOO1, the plant disturbance was white by 50 equal weights of an FIR filter while scaled to have unit output power, an was chosen to be white with unit power. The auxiliary system was first opera disturbance canceler on. The difference between the plant output and the outp was very small, having a power of 1.29( l o p .
Theoretically,the dynamic response distortion was zero with the disturbance ca ating. With the disturbance canceler turned off, the difference between the plan the output of M ( z ) was much larger, having a power of 2.97(
The difference between (9.23) and (9.22) indicates that the disturbance canceler reduce the dynamic response distortion by a factor of 23, or a reduction of 13.6
9.3 SIMULATION OF ADAPTIVE INVERSE CONTROL SY
FOR MINIMUM-PHASE AND NONMINIMUM-PHASE P
Adaptive inverse control systems have been simulated to control both the mini and nonminimum-phase plants that have beenzxperimented with in the previo These simulations did not use dither to obtain G ( z ) .The controller in each case by thz command input signal. The controller output was used by the adaptiv find G ( z ) . A block diagram of the adaptive inverse control system for the nonrnin plant is shown in Fig. 9.2. The block diagram for the minimum-phase plant except that the compensating network is omitted (it is not needed to stabilize the phase plant).
Figure 9.2
~
S
1 -2-I -
hiidei hold
I I
k = 24
(s
s+ I
+ 7)(s - 2)
Compensating network
7
(s
s -0.5
+ I)(s - I ) Plani
Plant disturbance
+ c
@I-+ I I
An integrated adaptive inverse control system without dither for nonminimum-phase plant.
II
Samples of plant output
9.4
SUMMARY
One fine result obtained in this chapter is the following: The effects of A P and with regard to distortion in the dynamic system response of the adaptive inve system of Fig. 9.1. This result is surprising and remarkable. In this chapter, we studied the control system of Fig. 9.1. Distortion in i caused by dither noise, uncanceled plant disturbance, and the dynamic effects AC. Since the latter cancel, the total plant output noise power is the sum of the d at the plant output and the power of the uncanceled plant disturbance.
[ total ) plant
\
noise power )
i=O
The dither power at the plant output is plant dither power
i=O
To use these formulas, one needs to have some idea of the plant impulse respo to be able to estimate C s p:. In addition, the power of the uncanceled plant is &c(O), given by (8.10). A discussion of how to estimate dcc(0) is given in where Eq.(8.10) is introduced. In Chapter 8, we obtained Eq. (8.41) which gave us a formula for optimal d This formula should be used when there is dynamic system error due to AP. stronger dither gives smaller A P but at the same time, more noise at the plant the system of Fig. 9.1, we are not greatly concerned about A P since the dyna error due to A P is canceled by the error A C in the controller. Therefore, Eq.(8 evant to the system of Fig. 9.1. To choose an appropriate level of dither power for the system of Fig. 9 ommend that the dither noise power at the plant output be made equal to the p uncanceled plant disturbance, which is irreducible. There is no point in makin power much smaller. Accordingly,
The terms are defined in Chapter 8 where these expressions are derived. How t the parameters is explained in the summary of Chapter 8. An expression for the time constant of the system of Fig. 8.7 has been o Chapter 8. This same formula gives the proper time constant for estimation o system of Fig. 9. I . The time constant is
This time constant determines the learning rate for the entire system because-ek the other variable parts of the system, are obtained essentially instantly from Pk processes. We now have completed our study of linear single-input single-outputi ad verse control systems. In the next chapter, we shall consider multiple-input multi adaptive inverse control systems.
Multiple-Input Multiple-Outp (MIMO) Adaptive Inver Control System
10.0
INTRODUCTION
All the systems described thus far have been of the single-input single-output (SIS The purpose of this chapter is to show how the basic ideas of adaptive inverse con be extended to apply to MIMO systems. The building blocks of such systems have functions that are matrices of transfer functions. Commutability of transfer functi is possible with linear time-invariant SISO systems is not possible with the more MIMO systems, since the order of matrix multiplicati%ncannot in general be com The offline methods used in previous chapters to find [Ck(z)] and [ Qk(z)] depend o mutability of transferiunctions, and therefore cannot be used with MIMO system ] be required. methods for finding [Ck(z)] and [ Q ~ ( z )will Before developing the techniques of adaptive inverse control for MIMO syst begin with a brief introduction to the basic concepts of linear MIMO systems and th resentation.
10.1 REPRESENTATION AND ANALYSIS OF MIMO SYSTEMS
Figure 10.1 shows a linear dynamic MIMO filter. Its array of K-inputs, after z-transf can be represented by the column vector [F(z)]. Its array of outputs, having the sam ber of elements as the input array, is represented by the column vector [ G ( z ) ] .The function of the MIMO filter is represented by a square K x K matrix of transfer fu H12(Z) ." H22(z) ...
HIK(Z) H2K(Z)
HKI(Z) HKZ(Z) . . '
HKK(Z)
Hll(Z) H21(z) [H(z)l = 270
The output vector can be expressed as [G(z)l= [ H ( z ) l [ F ( ~ ) l .
Each output is a linear combination of filtered versions of all the inputs. The tr tion from input j to output i is Hij(z). A schematic diagram of [ H ( z ) ]is shown in Fig. lO.l(a). The signal path line j to output line i is illustrated in Fig. lO.l(b). A block diagram of the M is shown in Fig. lO.l(c). The input vector is [ F ( z ) J .The output vector [ G ( z [ H ( z ) ] [ F ( z ) ]The . overall transfer function of the system is [ H ( z ) ] . Other configurations of MIMO filters are shown in Fig. 10.2. Filters [ H 2 ( z ) ]are in parallel in Fig. 10.2(a). The input signal for this system is [ F ( z ) l [ H 2 ( z ) ] [ F ( z ) ]The . outpu signal is the sum of two signals, [ H l ( z ) ] [ F ( z ) and H 2 ( z ) ] [ F ( z ) ]The . transfer function of this system is therefore equal to [ H ( z ) H 2 ( z ) ] .In Fig. 10.2(b), [ H I ( z ) ]and [ H ~ ( z )are ] in cascade. Following signals system yields certain facts. The signal at the input node A is [ F ( z ) ] ,and the si B is [ H l ( z ) ] [ F ( z ) ]The . signal at the output node C is [ H z ( z ) I [ H I ( z ) ~ [ F ( function of this system is therefore equal to [ H ( z ) l = [ H ~ ( z ) I [ H( zI ) ] . This is of the matrix transferfunctions, in reverse order to the signalflow. In Fig. 10.3, a system with a feedback self-loop is shown. To find the tran of this system, we note that the output [ G ( z ) ]can be expressed as [ G ( z ) l = [ F ( z ) l + [HI( z ) l [ G ( z ) l .
J Figure 10.3
A
MlMO feedback loop.
Premultiplying both sides by the inverse of [I - HI(z)] gives
[G(Z)J= [I - HI(z)J-'[F(z)I. The transfer function of the system is therefore
[H(z)I = [I - HI (d1-I.
Another MIMO feedback system is shown in Fig. 10.4. This system can be red the transfer function by following a number of steps, illustrated in Figs. 10.4 and 10.4(c). The original system is shown in Fig. 10.4(a). The input is [F(z)] . diagram is redrawn in Fig. 10.4(b),with the same input caus put is [ G ( z ) ] The output. The self-loop is the cascade of two branches whose transfer functions a in reverse order in Fig. 10.4(c). The overall transfer function may now be obt spection, since the system is reduced to acascade of a self-loopand a branch [H transfer functions of this cascade in reverse order, the overall transfer function
[H(z)1 = [HI(Z)I[I- H2(z)H1(z)l-'.
The same system can be reduced in another way, as illustrated in Fig. 10. inal system is shown in Fig. 10.5(a). A step in the reduction is shown in Fig. 10 again, the input [ F ( z ) ] produces the output [G(z)]. The self-loop is simpl 10.5(c), and the transfer function can be written by inspection as
[H(z)l = [I - Hl(2)~2(z)l-"Hl(z>l.
[F(z)l
+GJ [Hi(z)l
[H~(z)I[HI (z)l
(C)
Figure 10.4 Reduction of a MIMO feedback system: (a) Original system; (b) Equivalent system; equivalent system.
This result should be the same as (10.7), that is,
Equation (10.9) is an identity, as the reader can easily verify. One more example will help to solidify our understanding of MIMO syste 10.6 shows a system with two feedback loops. The original system is shown in F The two feedback loops become self-loops in Fig. 10.6(b)without any changes t output transfer function. The self-loops are simplified in Fig. 10.6(c), and th functions are summed in Fig. 10.6(d). From here, the transfer function can be inspection as
With this brief introduction, we are now prepared to develop adaptive inve for MIMO systems.
(C)
F i p r e 105 Reduction of a MIMO feedback system by another approach: (a) Original system; system; (c) Simplified equivalent system.
ADAPTIVE MODELING OF MIMO SYSTEMS
A MIMO plant to be modeled may be represented by the following transfer fun
The plant input is an array of signals whose transform is represented by the vec
Figure 10.6
Reduction of a MIMO feedback system with hv0 h p S : (a) Original system; (b) Equiv
(c) Simplified equivalent system; (d) Further simplified equivalent system.
10.2.1 Adaptive M I M O Modeling Using Scheme B
The plant can be adaptively modeled, using independentdither inputs, as shown input two-output case of Fig. 10.7. Use of uncorrelateddither in the modeling pro essary when the plant inputs alone cannot be used, that is, when they are correlate other or when they are not persistently exciting. The dither scheme illustrated is a MIMO version of scheme B, which is shown in Fig. 4.3(b) for the SISO ca succinct picture of the dither scheme is shown in Fig. 10.8, making use of a v diagram. The modeling grocess igustrated in Fig. 10.7 involves the adaptation of rate filters, 61, P21, and P22. As these filters converge, their transfer funct closely approximate the respective plant transfer funztions PQ Pl2, P21, and P2 a common error signal is used in the ad_aptation_of PI1 and P12, and that anoth error signal is used in the adaptation of P21 and P22. It seems that the adaptation these filters should be affected by their interconnections, and a simple analysis is indeed the case. Of concern are issues such as stability of the adaptive proce convergence, misadjustment, and noise in the weights. Figures 10.9, 10.10, 10.11, and 10.12 have been devised as an aid in addr issues. Figure 10.9 shows one-half of the adaptive process illustrated in Fig. 10. operates independently of the other half and can be studied all by itself. The n ing signal, the controller output, is not used for plant identification by scheme B signal serves this purpose. In fact, the natural driving signal acts as noise to t identification process. Figure 10.10 is equivalent to Fig. 10.9, except that her ference due to the controller output is accounted for by an additive noise comp plant output, and it appears in addition to the plant disturbance. Assume that all of the filters of Fig. 10.10 are FIR and that they all are time samples. Assume that the two dither signals of Fig. 10.10are white and of e Figure 10.11 shows how a single white dither input could be used under these cir instead of the two dither inputs of Fig. 10.10. Two dither signals are created fr one by using a delayzf n ti%e samples. In this way, all of the signals at the tapped delay lines of PII and P12 will be mutually uncorrelated in Fig. 10.11just in Fig. 10.10. The results of adaptation in both systems will therefore be the sa TJe syste? of Fig. 10.11 can be redrawn as shown in Fig. 10.12. The tw filters P II and P12 are now combined to comprise a single adaptive filter of len@ samples. Assume for simplicity that the values of I* are set to be the same for P
62,
r Laif,
disturbance roller Put
-++ +
I
I I
Dither input #
I
I
I+ +
Error
I Dither input # 2
-
L
r
I Figure 10.7
6 2(Z )
t Adaptive model
[&(r)l
I
Modeling a two-input two-output system in accord with scheme B.
Dither input vector
Figure 10.8
Simplified block diagram for modeling a MIMO plant based on scheme
Plant disturbance
Controller output
Dither
Figure 10.9
A part of the two-channel MIMO modeling process.
4
h(t)
I +
Dither
FI*(Z)
Figure 10.10
A part of the two-channel MIMO modeling process, simplified.
The issues of stability, and convergence rate, and misadjustment and n weights are the same for the system of Fig. 10.12 as they are for the system of Fi der the conditions assumed above. Note that the adaptive behavior in Fig. 10.9 as that in Fig. 10.7. So, to understand the adaptive behavior of the system of Fig needs only to analyze the behavior of the system of Fig. 10.12. This can be don isting methodology. Formulas for time constant, stable range of p, and_mis?dju given in Appendix B. Since the length of each of the adaptive filters P11,Pl2, Fig. 10.7 is n, then the length of the equivalent adaptive filter in Fig. 10.12 is tw more channels are involved, and the number of MIMO channels is represented the equivalent filter length is n multiplied by K. Using formulas (B.20) for time constant, (B.22) for stable range of p , and misadjustment, the corresponding formulas for scheme B applied to a multichan system are
0
1 n K E [a:]
’
PlZ(Z)
’
Delay of n time units
Figure 10.11
Alternative dither for two-channel MIMO modeling process.
Everything stays the same, except that the equivalent filter is long for MIMO.T adjustment down at a low level, it is necessary to adapt more slowly, roughly equal to K.
10.2.2
Adaptive M I M O Modeling Using Scheme C
As scheme C was often the preferable choice for SISO systems, it is of course i consider its application to MIMO systems. Figure 10.13 shows a detailed plan channel MIMO application of scheme C. Figure 10.14 gives an overview vec for general use of scheme C for plant modeling in a multichannel MIMO syste Figure 10.15 shows half of the two-channel modeling modeling process of Since it and the other half act independently, adaptive behavior can be determ study. We attempted to relate the behavior of the system of Fig. 10.15 to that dimensional SISO system of scheme C (shown in Fig. 4.3(c)), but this did no Such an approack worked out weJ for scheme B above, but failed for scheme C the presence of PI1 COPY and P12 COPY. The various filters cannot be stacke make longer filters in this case. To analyze the system of Fig. 10.15 in order to gain an understandingof th Fig. 10.13, it is necessary to go back to fundamentals. We will use the analytica developed in Section B.4.
Figure 10.12
Detailed diagram of two-channel MIMO modeling with alternative dither.
Controller output
-t" Dither input# 1
Error # 2
Dither input # 2 Adaptive moder[&(r)]
Fipre 10.13
Scheme C for two-input two-output plant modeling.
[P(z)]
Error
Dither
Figure 10.14
A vector signal diagram of scheme C for MIMO plant modeling.
Referring to Fig. 10.15, we observe that the average mean square error mum mean square error, equal to the plant disturJan5e powerglus the averagze , 9 1COPY and Pi2 square error caused by noise in the weights of PII , P I Z and The average mean square error can be written as
g = emin + (average excess MSE) = tmin
+ w E E [ ~ +~ L~ LI ~ ~ E [ S & I + CLniE[(U;k)21.
+CLnimJ"Zl
If we assume that the dither power is the same on both channels, and if we assu controller output power is the same on both channels,
t = {min + 214
[E[$I+ E[(u;)'I].
Note that E[S;] is the dither power on one channel, and E [ ( u ; ) ~is] the contr power on one channel. The average mean square error can be written as tmin
+
= 1 - 2pn [E[S,2] E[(u;)2]]
-
Smin
1 - 2pnE[ui] *
The expectation E[u:] is the power level on a single-input channel to the plant. These expressions can be generalized for the multi-input case. Assume ag dither powers are equal from channel to channel, and that the controller output equal from channel to channel. The number of channels is designated by K . Ac emin
= 1 - pnK [ E [ $ ]
+ E[(u;)2]]
Disturbance
++
Controller output
UZk
+
Error Dither input# I
Slk _____f-
p71(2)
I Dither input # 2
Figure 10.15
P72(2)
A part of the two-channel
MIMO modeling process in accord with schem
of p for MIMO scheme C is
1 >p>o. KnE[u:]
If the controller output power differed from channel to channel, and/or the dither fered from channel to channel, new expressions for misadjustment and stable r be derived using similar analytical techniques.
10.2.3 Conclusions on MlMO Plant Modeling
Comparing the expressions for misadjustment and stable range of p obtained for tems and given in Appendix B with the correspondingexpressions for MIMO sys in this chapter, one can draw a simple conclusion. The SISO expressions can b ized directly to apply to the MIMO case by simply multiplying the number of w adaptive filter n by the number of channels K provided that, for the MIMO syst a. all filters have the same number of weights; b. all filters have the same input power levels: c. dither powers are equal on all channels; d. the same value of p is used for all adaptive filters.
This generalization works for systems based on both scheme B and scheme C. It is interesting to note that for a given level of misadjustment, convergen creases only linearly with the number of MIMO channels. Before doing this a were concerned that learning time would be proportional to the square of the channels (since the total number of weights grows with the square of the numb nels). Fortunately, learning time is only proportional to the product of the numb nels and the number of weights per adaptive filter. This is a surprising result.
ADAPTIVE INVERSE CONTROL FOR M l M O SYSTEMS
So far, several ways for adaptively modeling a MIMO plant have been descri chapter. Having a plant model [ P ( z ) ] ,one can use it to develop an inverse co the plant [ P ( z ) ] .Two ways of finding the inverse controller will be considered way is primarily an algebraic technique, and the second way is a technique ba filtered-r approach to adaptive inverse control.
] the MIMO system would be that of the signal flow,a proper [Ck(z)for [~(zolr~k(z)l= [M(z)I.
An algebraic technique for obtaining [ck(z)] is the following. Refer to The matrix [&I COPY can be obtained by the methods illustrated in Figs. 10 A new matrix transfer function [Vk(z)] is introduced here for mathematical pu adaptive process for finding [VA.(Z)]is indicated in Fig. 10.16. From [vk(Z)], a process can be used to find the controller [Ck(z)].
Ar
Ar White noise vector
b I
/
[Vk(Z)l
/
+ Figure 10.16
A step in the calculation of [I%)].
Assume that the reference m$el [M(z)] is chosen to give good system op that its inverse is stable. Then, let [Ck(z)Ibe = [M(z)l-"Vk(z)l[M(z)I.
[?k(Z)I
Justification for Eq. (10.26) will be_given below. Using Eq.(10.26) to obtain [ck(Z)], we next show that (10.25) will be sati to Fig. 10.16. Assuming that the adaptive process has converged and that th small. Then [Vk(Z)l[&(Z)l = [M(z)I. Postmultiplying both sides by
[6(z)]-' yields
[Vk(Z)I = [M(z)l[&(z)I-'.
Substituting into (10.26) yields [Fk(Z)l = ~
=
M ~ z ~ l - l ~ ~ ~ z ~ l ~ & ~ z ~
r 6(z)l-' [M(z)l.
[M(z)] is chosen by the system designer, it should be chosen so that [M(z)]-' i This restriction can be relaxed if a sufficiently delayed stable inverse o used.] Referring to Eq.(10.26), the delay incorporated in [M(z)]-' could be co for by using a correspondingly time advanced form of [M(z)]. This can be readi if the reference model [M(z)] has sufficient transport delay in all of its impulse Without sufficient transport delay in [M(z)], one could simply redefine [M(z)] t delay. The end result would be an overall system response that would be like a de of the original [M(z)]. Analysis of the performance of adaptive inverse control for MIMO system in like manner to that for SISO systems as explained in Chapter 6. The overall sy consists of a sum of four components: i. Plant output disturbance ii. Dither noise filtered through the plant iii. System error due to truncation of [ ?(z)] andor [c^(z)I iv. Dynamic system error
Calculation of noise at the plant output due to plant disturbance ankdither noise forward. The effects of system error due to truncation of [ P ( z ) l or [C(z)l can be m by making the involved adaptive filters sufficiently long. We assume that trunca arr negligible. The dynamic system error is dueLo the effects of noise in the [C(z)], originating from noise in the weights of [P(z)]. This component of error can be calculated in an analogous fashion to the given by Eqs. (5.26)-(5.41)and Eqs. (6.6H6.11). From Eq. (10.30), onecan w
[E(z)l = [?(z)I-"M(z)l.
It is required that [?(z)]-' be stable. For this to be, it may be necessary to mak a delayed inverse. Continuing,
[&)I
= [Pl(z)l-'[M(z)I = [P(z) AP(z)l-'[M(z)l
+
= [[P(z)I(I
+
+ [p(z)l-'[Ap(~)l)]-'[M(z)l
= (I [P(z)l-'[Ap(~)l)-'[P(z)l-'[M(z)l (I - [ P ( z ) 1 ~ ' [ A P ( z ~ l ) ~ P ~ z ) l ~ ' ~ ~ 'It can be shown that the delayed inverse of any [ M ( z ) ]will be stable, with enough delay.
Equation (10.34)is completely analogous to the SISO Eq. (5.37).Figure 5.15sh SISO case, how the dynamic system error forms as the command input signa through the system and encounters the weight noise AP(z). For the MIMO case shows the formation of the dynamic system error.
Figure 10.17
MIMO dynamic system error due to fluctuation in [p(z)].
The variance of all components of the dynamic system error vector can b
in an analogous way with a method like the SISO calculation. A simple formul to the SISO formula in Eq.(5.40) can be obtained if special conditions exist, su
a. The power levels on all channels at point B in Fig. 10.17 are the same. Th alent to the requirement that the power levels on all output channels of th be equal.
b. All filters of [p(z)]have the same number of weights n. All filters of [?(z same number of weights n. c. The values of 1 for the adaptive filters of [ p(z)]are equal. d. The plant disturbance power levels are the same on all channels.
At the plant output, all channels have the same dynamic system error variance. plant output channel,
f varianceof )
[
power
This relation is analogous to Eq.(6.6), and all definitions made in connections wi relevant here. Other special assumptions are necessary for (10.36) to apply, and e. Dither power is equal on all channels.
f. All the filters of [ F ( z ) ] have essentially equal sum of squares of the impul impulse responses.
For the special case above, simple relations can be ob5ined for optimal dit With a given choice of time constant r for adaptation of [ P ( z ) ] ,the overall sy power for a single-output channel can be expressed as
This equation is based on the SISOJormula of Eq. (6.6), and all definitions ma to (6.6) apply here. Adaptation of [ P ( z ) ]is done by scheme C. Differentiating(1 respect to E[S;] and setting the derivative to zero yields the optimal dither powe
This equation is analogous to the SISO relation of Eq. (6.9). The minimum overall system error power can be obtained by analogy to E Using the optimal dither with scheme C, the result for a single-output channel is
10.3.2
The Filtered+ Approach
Adaptive inverse control of MIMO systems can also be done by making use of th method of adaptive inverse control introduced in Chapter 7. None of the other a verse control techniques taught above, including the filtered-X algorithm, can be
] , error filter, are indicated in Fig. 10.18. The p means for finding [ P ~ ’ ( z )the [ p ( z ) ] is , obtained using scheme C. The method is analogous to that described 7, Figs. 7.4 and 7.5, for SISO systems. A few details about finding the delayed plant inverse filter, especially p MIMO systems, need to be discussed. Suppose that online inverse modeling is (switch left in Fig. 10.18). The inverse modeling signal is then the filtered erro indicated in Fig 10.18. The reason for using the filtered error for this purpose the input for [Pi’] to have the right spectral character. Since a copy of this f having ck, the overall system error vector, as its input while generating the filte is reasonable to have the same or approximately the same input signal or a sig equivalent spectrum for an input as this filter is created by adaptation. Having th spectrum during adaptation is not critical, but taking the trouble to “do it right” a better error filter. Offline adaptation (switch right in Fig. 10.18) could be ac by using synthetic noise generated for inverse modeling whose spectrum shoul to match that of the filtered error as well as can be done. Offline adaptation wou be done at the outset, to initialize online adaptation. On the other hand, offline is an excellent idea and it can be used in its own right. Analysis of the system of Fig. 10.18 has been attempted, but because of t mutability of the Eatrix operator, no simple expressions for misadjustment, . r a n g of p for stability have-een obtained so fa weight vector of [ C ( Z ) ]or constants for the adaptation of [ C ( z ) ]are the same as if [ C ( z ) ]were isogted f of the system. The time constants for the adaptation of the plant model [ P ( z ) ]b are determined by the met&& of Section 10.2.2. The time constants for the ad the inverse plant model [ P i ’ ( z ) ]are the same as if the subsystem for calcula isolated from the rest of the system.
10.4
PLANT DISTURBANCE CANCELING IN M I M O SYSTE
Plant disturbance canceling in SISO systems was described in Chapter 8. The niques can be utilized, with some modification, to cancel disturbance in MIMO p ification is required in some places in order to ensure that the ordering of signa disturbance canceling feedback is not commuted. Figure 8.1 shows a SISO plant disturbance canceler using online adapta Q ( z ) , and Fig. 8.3 shows a SISO plant disturbance canceler using an offline proc Q ( z ) . Disturbance cancelers such as these should generally not be used for MIM tions. The reason is that the plant disturbance must first be filtered by Q ( z )and t to P ( z )to achieve disturbancecanceling. But Q ( z ) is formed in the systems of F
3 , I+ I
Command input vector
.......... C(Z) I-.: ..........
.
/ * \
C_OPY
-
[P(z)]
v
i 4
Plant
disturbance
L +
-
COPY Filtered error vector
+
~~
Figure 10.18
Ek
[p?;'(z)]
Reference model
Overall system errorvector ,
Mz)1 ~~
Filtered-t adaptation for a MIMO plant.
+
of Fig. 8.3. Online processes can be devised alternatively, and the resulting sy be analogous to that of Fig. 8.1. It may be noted that the use of a delayed inverse in Fig. 10.19 often make verse and otherwise causes no additional problems. The delay must be accoun ever, when adapting Q ( z ) . This has been done by delaying the input of [ Q ( units of time, A units for the delayjn the inverse of [ P I , plus one unit for the cascade [ Q ( z ) ]COPY,z - ' I , and [ P ( z ) ]COPY. Referring again to Fig. 10.19, we should note that the synthetic noise ve one used to obtain [ Q(z)] should, in principle, be spectrally like the plant distu spthetic noise vector number two used to obtain [pi']should be such that [ P i ' ] would be spectrally like the error signal of the process for generating [ Q having this kindEf spectrum is not so easy to generate. White noise should suffi since errors in [ P i ' ] are not critical for finding the best [ Q ( z ) ] ,as we learned case. It is useful to note that if [ P ( z ) ] is minimum-phase and if the sampling high so that z-' is a very small delay, then the error in generating [ g(z)]will b and [ Q ( z ) ]will be a close approximation to an exact inverse of [ P ( z ) ] .Whe case, [ Q ( z ) ] and [ p(z)]are commutable and the methods of both Figs. 8.1 an be directly applicable as alternatives to the methods developed in this chapter if to use them.
10.5
SYSTEM INTEGRATION FOR CONTROL O F T H E MIMO P L A N T
Figure 9.1 shows an integrated control system for a SISO plant. A modified ve system is shown in Fig. 10.20 for a MIMO plant. The disturbance canceling techniques of Fig. 10.19 have been incorpora 10.20. What is also added is an offline process for finding the controller, and thi the filtered error algorithm. Notice that in all cases, the adaptive offline proces figured so that the training signal flows occur in the same sequence as in the ac This is necessary for MIMO. Another method for finding the controlLer based on filtered error is sh 10.21. Offline processes are used for finding [Pi'] and [ Q ( z ) ] .The process fo controller is online. The result is a system that learns a little more slowly than of Fig. 10.20 but would provide the correct controller for the plant and its noi system even if [P^(z)]does not perfectly represent [ P ( z ) ] .
hetic vector1
Input vector
P COPY
[Qt(z)l /
I
Figure 10.19
z-'I
#
I
C_OPY
[pk (Z)]
t
4
t
-
I I I I I I I 1 I I I I I I
Unit delay
*
Synthetic noise vector2
I
~opy [4(z)]
b
Control system output vector
-
MIMO plant disturbance canceling using offline formation of [Pi'] and [ Q ( z ) ] .
I I
I
4 COPY [ck(Z)]
/ Controller
I
Figure 10.24)
I
\
button
/
COPY /
[fk(Z)]
#
An integrated MIMO system.
.
I I
I
*
Plant
output vector
I I
(z) command l vector
4 COPY
[ck ( Z ) ] I Controller
! [Cdz)]
I
I A
Another integrated MIMO system.
Reference model
Figure 10.21
1
-
I
Plan1
OUtpUl vector
ger compartment of commercial aircraft. A new generation of airplanes is being using a combination of turbojet engines and propellers that promises to be 30 to more fuel efficient than the best turbojet airplanes flying in the mid 1990s. The e pusher propellers will most likely be mounted at the tail. A serious drawback is in cabin noise resulting from the use of propellers. We propose to utilize adap control systems of the MIMO type to control and cancel noise due to the turbo and the propellers. Referring to Fig. 10.22(a), we see the aircraft with a turbine-driven prop rear. We assume that there will be two engines and two propellers, and that t synchronized. Thus, obtaining a single reference signal from a sensor on the sha the engines should be sufficient. We will need to obtain signals corresponding damental and harmonics of the turbine blades, and the fundamental and harm propeller blades. A single reference signal containing a sum of all those compon be satisfactory. The shaft sensor signal might need to undergo nonlinear proces erate all the important harmonics. The proposed approach to the problem of canceling the noise is presen 10.22(b). Each passenger seat has an internal microphone located approximatel senger’s head level. Loudspeakersare placed inside the aircraft cabin at some dis from each other and from the microphones. The engine reference signal is fed to noise canceler. The microphone signals are also fed to the adaptive canceler. Th the canceler is a set of loudspeaker signals, obtained by optimally filtering the e ence signal and intended to drive the loudspeakers so that their acoustic outputs the ambient engine and propeller noise in the vicinity of the microphones, nea of the passengers. The system illustrated uses two microphones and two loudsp number of microphones and loudspeakers could be increased as required. The objective for the adaptive system is to generate the loudspeaker sign to minimize the sum of the powers of the microphone outputs. The microphone the error signals of the system. Figure 10.23 is a schematic diagram showing the the propagation paths of the engine and propeller noise and the loudspeaker so Each microphone senses the sound, a sum of these components, and outputs a signal in accord with its acoustic to electrical transfer function. Adaptive filte loudspeakers. Their inputs come from the common engine reference signal. T is: How should one adapt the adaptive filters? An answer to this question comes from the block diagram of Fig. 10.24. phone signals, the error vector of the system, are minimized in the mean square s mize the mean of the sum of the squares of ikcomponents) by using the filtered+ of the MIMO type to adapt the controller [C]. To filter the error, we need a de inverse. The way to get this inverse is shown in Fig. 10.25.
Passenger compartment
Baggage Compartment
Figure 10.22
J
Aircraft noise problem and adaptive system for its mitigation
Figure 10.23
h
F
Schematic of noise propagation paths and noise canceling system.
Mic. signal # 1
Mic. signal # 2
I
Figure 10.24
co;troller
Dither vector
Loudspeaker signal vector
/
-
. ,
+
Error vector
funct.
Engine and propeller noise at passenger seats.
+ Seats
Loudspeaker sound at passenger
- u
[El
I
Plant inverse Vector block diagram of adaptive aircraft noise-canceling system.
Microphone signal vector
Engine and propeller noise at passenger
Loudspeaker signal vector
Loudspeaker sound at passenger seats
k
+
I
\
funct. COPY
'
Dither
r"""""""""'-,,,,------~ I
I I
Synthetic noise vector
I I I I
COPY
Plant inverse
I I I
I I I
Offline process
~,-,,,,,,,,,,,--,----,----,,,-----
Figure 10.25
Plant modeling and inverse modeling for the aircraft noise-canceling sys
The noise canceling system diagrammed in Figs. 10.22-10.25 uses adap control techniques to control noise for the passengers in an aircraft. Once again
microphones would be treated like plant noise, and the loudspeaker outputs wou to cancel this noise. Many other ways can be thought of to approach and solve this noise canceling problem. The First InternationalConference on Active Control of Sound and Vibratio at Virginia Polytechnic Institute, April 15-17,1991. The meeting was sponsored Langley Research Center, Office of Naval Research, and the U.S. Army Aerostr rectorate. The Conference Proceedings were published by Technomic Publishin (85 1 New Holland Ave., Box 3535, Lancaster, PA, 17604,USA, Fax: (7 17)295-4 Proceedings were edited by Professors C.A. Rogers and C.R. Fuller of Virginia P Institute and State University, Blacksburg, Virginia. Papers were presented descr on the above problem by British Aerospace and others. Many papers reported on fan noise in air conditioning ducts, canceling auto exhaust noise, and canceling in cars. Most of these projects make use of the filtered-X LMS algorithm. Ada and vibration canceling has become a major new field.
SUMMARY
In this chapter, means of describing linear multiple-inputmultiple-output (MIMO have been developed. Block diagrams and flow graphs are useful for this purpose gebraic methods. Adaptive techniques for modeling and inverse modeling were i and they turned out to be very similar to those used with single-input single-out systems except that care is exercised not to commute matrix transfer function op Formulas for misadjustment and time constant of the adaptive MIMO plan process have been obtained when using dither schemes B or C. For a given level justment, learning time is the same as for SISO, multiplied by the number MIMO K. It is a surprise that learning time goes up only linearly with K ,not with K 2 fo Inverse controls for MIMO plants were devised. One approach was based gebraic technique. A second approach was based on the filtered+ LMS algori methods work quite well. Cancelation of plant disturbance in MIMO systems is possible. Several met explained for this, offline and online. The filtered-c algorithm proved to be quit fizding the disturbance-canceling feedback transfer function [ Q ( z ) ] from the pl [ P ( z ) ] .Care was taken in all these developmentsto ensure that the ordering of ma fer functions was not commuted. Adaptive inverse control systems were iescribed for MIMO plants. Tw approaches for finding the inverse controller [ C ( z ) ]were demonstrated, both ba filtered-c algorithm, one offline, the other online.
tems is a strange one, because nonlinear systems do not generally have inverse linearity invokes even more rules. In the next chapter, we develop techniques li inverse control for application to nonlinear SISO and MIMO systems.
Nonlinear Adapti Inverse Contr
11.0
INTRODUCTION
The principles of adaptive inverse control can be applied to the control of nonlinear The purpose of this chapter is to show how to do this using SISO plant control as an exa Nonlinear MIMO plants can also be controlled with systems of greater complexity. Many of the rules of MIMO systems apply to nonlinear systems, such as no mutability of filtering and signal processing operations. Additional rules apply to non systems, such as exploring or modeling plant behavior only with input power level and signal characteristics set to correspond to that of the actual plant input. Scaling and lin do not work. Strictly speaking, nonlinear systems do not have inverses. Nevertheless, the m of adaptive inverse control can be made to apply. Inverse control of nonlinear plan be done and canceling disturbance in nonlinear plants can also be done by making nonlinear adaptive filters. For inverse control, the command input is applied to a nonlinear controller who justable parameters are adapted so that when the output of the controller drives the input, the plant output becomes a best least squares match to the reference model’s o The resulting controller would be a good inverse only for the particular input comman nal, not in general. If the characteristics of the command input signal were to cha would be necessary for the controller to adapt rapidly and keep up with the change long as this is feasible, nonlinear inverse control will work. Let this be the case. Ou step then is to study nonlinear adaptive filtering.
11.1 NONLINEAR ADAPTIVE FILTERS
One of the simplest of nonlinear adaptive filters is shown in Fig. 11.1. The input sig applied to a tapped delay line. Signals at the taps are weighted, squared and weighted, forth, with all weighted signals summed and output. The output signal is therefore a combination of the signals at the taps plus a linear combination of squares of these si The desired response input is compared with the filter output signal. The differenc
Another way to construct an adaptive filter is illustrated in Fig. 11.2. The is applied to a tapped delay line, and the tap signals are in turn applied to an ada network. A three-layer network is shown in the figure. Each neural element, labeled “AD,” is an independent nonlinear adap called Adaline. Connected into a network, these devices allow the realization nonlinear adaptive systems. The neural device is an adaptive threshold elemen studied by Rosenblatt [ 11 and by Widrow and Hoff [2] in the early 1960s. An a ear bypass is shown in Fig. 11.2, which contributes to an output which is a com linear and nonlinear functions of the tap signals. A method for adapting multilayered feedforward networks as in Fig. 11.2 oped in the early 1970sand 1980s. The algorithm is called backpropagation[3-5 explanation and derivation is given by Widrow and Lehr [6] in the Septembe of the Proceedings ofrhe IEEE.‘ The September and October issues of these P are special issues devoted to neural networks. General references on the sub ral networks are the IEEE Transactions on Neural Networks, the journal Neura published by the International Neural Network Society (INNS), and the Journ Computationpublished by M.I.T. Press. Many internationalconferences on neur are held in the United States, Europe, and Asia each year. Their published proc always a source of the latest information on the subject. Many books have also b on neural networks. A representative sample is given by references [5,7-131. U networks for nonlinear control has been proposed by Kawato, Uno, Isobe, and S and by Psaltis, Lideris, and Yamamura [ 151. Their works relate well to the su of this chapter. A collection of papers on this subject has been compiled by Mi and Werbos [ 161. Backpropagation is a very sophisticated generalization of the LMS algor an instantaneous gradient with the method of steepest descent to minimize mea ror. The mean square error is not a quadratic function of the weights, howev known nature of the mean square error function makes prediction of rate of c very difficult. Furthermore, local optima exist and often the backpropagationalg hang up on local optima. This does not happen with the Volterra filter, whose simpler and in many respects, more predictable. How to configure nonlinear adaptive filters cannot yet be determined by a cedures. Present day methods are empirical. There is no simple way to decide i filter or a neural network filter would be the best for a given application. If po ‘Thispaper is reproduced here as Appendix G.
I Figure 11.1
Output
. Desired response A simple nonlinear adaptive filter.
-
ayer
ayer
ayer
Figure 11.2
I
+ An adaptive nonlinear filter based on neural networks.
Desired response
* Error
added to the plant input signal. This is like scheme A of Chapter 4,and since w n_onlinear modeling, let this process be called scheme ANL. The input to the ada P is the same as that of the plant. This is as it should be when dealing with a plant. Modeling schemes B and C of Chapter 4 should be avoided since when u the dither signal inputs to the adaptive modeling filters are typically small in am are therefore quite different in character from those going into the respective pla nonlinear modeling algorithms like schemes B and C will be described below. Plant disturbance
Dither
Plant input
*
adaptive filter
Figure 11.3
Modeling a nonlinear dynamic system using scheme A N L .
When doing modeling in accord with scheme ANLof Fig. 11.3,care must b utilizing the dither signal. If the plant input is stationary, the dither should not be u plant input is nonstationary,dither can be used having the intensity and character t adaptive filter to model the plant under input conditions of the greatest interest. A time, one should keep the dither as small as possible to minimally disturb the plan technique that could be used to achieve these objectives would turn the adaptive off and on to cause the adaptive filter to model the plant under selected conditio
11.2.1
A Nonlinear Modeling Scheme like Scheme B
Using the simple adaptive Volterra filter that is illustrated in Fig. 11.1, one can means for modeling a nonlinear plant P that works very much like scheme B of Refer to Fig. 11.4. The nonlinear plant is represented for this illustration as a fix
Controller output
-
-++ 4-
Nonlinear plant P
1 A Copy weights "&l
I'
4-2
2-1
at
Dither
I
Error
Derivative model P^,
I
Figure 11.4
Scheme B N L for nonlinear plant modeling.
The adaptive derivative m d e l P' has 8 k as its input, but to produce an o to the response of the plant to 8 k , the adaptive derivative model must have acces cordingly, the adaptive derivative model has both u; and 8 k as its inputs. Its outp match the total plant output, just the component of the plant output due to 8 k . The between the plant and adaptive derivative model outputs is used as an error sign adaptation of the weights of the adaptive derivative model. This is the same sit encountered with scheme B. When the weights of the adaptive derivative model p^l correspond exa weights of the plant, the output of the model corresponds almost exactly to the of the plant output response due to the dither 8 k . The correspondence of these o ponents becomes exact in the limit as dither power is made small. An explana this works can be developed by referring to Fig. 11.4. The effects of small dither on the plant output at any moment of time can b by perturbing one at a time the signals at the taps of the plant's tapped delay lin and i l k - 2 . The derivatives of the output with respect to these signals multiplied by sponding dither components S k , & - I , and 8k-2 are summed to yield the plant ou component due to the dither. The adaptive derivative model P^, is structured to make an analogous co It does so when its weights correspond to the weights of the plant. Then, the ou derivative model is identical to the dither output of the plant. Making the wei derivative model equal to those of the plant causes the mean square of the err the outputs of the plant and the derivative model to be minimized. In fact, whe tive process adjusts the weights of the derivative model to minimize mean squar adaptive process causes the derivative model weights to converge o," the plant w call this adaptive algorithm scheme B N L .The derivative model P' is not a m plant. 15s merely used to find the weights of the actual plant, shown in Fig. 11.4 model P is a true Volterra model resembling the plant. Scheme BNLworks well for higher order Volterra plants and adaptive pla The simple Volterra form illustrated in Fig. 1 1.4can be generalized for plants hav degree nonlinearities and crossproducts of higher degree. A simplified block di provides an overview of scheme BNLfor modeling higher-order Volterra plants i ~ be used with adaptive plant models based on Fig. 1 1.5. Scheme B N L Calso works, utilizing adaptive filters of the type shown in Fig. 11.2.
11.2.2 A Nonlinear Modeling Scheme like Scheme C
Scheme B N Ladapts by matching the output of the adaptive derivative model to of the plant P. The output of the adaptive derivative model, at convergence, co
Error Figure 11.5
Scheme B N L .
the plant output component due to the dither input and does not contain the plan to u;. The error will therefore be large, even at convergence. The original sch fers from the same problem, and this motivated the development of the origina A scheme for nonlinear plant modeling which is analogous to scheme C is i Fig. 11.6. We call this scheme C N L .
Controller ouput
+
fl \
4
f Plant model
F /
‘t--+==~t-Derivative model
sk
Dither
7 I Error
Figure 11.6
Scheme CNL.
In Fig. 11.6, the nonlinear plant P can be considered to be of a high-or type. The plant model P^ is a Volterra filter with adjustable weights. The adap tive model is the same as the one shown in Fig. 11.5, and it is represented in
bance signal at convergence. It should further be noted that both schemes BNLand C N Lconverge on the lution even if there is plant disturbance and this disturbance is correlated with the output u;. The same is true for schemes B and C. This would not be true for sche for scheme A. This issue becomes important when developing methods for canc nal plant disturbance. Although not illustrated here, nonlinear adaptive plant models based on works can be implemented by means of the adaptive filter of Fig. 11.2 and its a derivative model. Schemes A N L , BNL,and CNLcan all be realized with neural n
11.3
NONLINEAR ADAPTIVE INVERSE CONTROL
The inverse controller for a linear plant has a transfer function which is a close tion to the reciprocal of the plant transfer function. In the nonlinear case, transfe do not exist. So what does inverse control mean for nonlinear plants? Control of a nonlinear plant can be achieved by filtering the command i with a nonlinear filter and by applying the filtered command signal as the input t The idea is the same as for control of the linear plant. We define this as inverse both linear and nonlinear plants. A model-reference adaptive inverse control system for a nonlinear @ant Fig. 11.7. The plant is represented by P and the adaptive plant model by P. Th model M could be linear or nonlinear. A simple plant modeling process is sh figure. In its place, schemes A N L ,B N L ,or CNLcould be used if dither is requir modeling. A-copy of the plant model P^ is used in an offline process to find the bes c^ and P are nonlinear, their ordering cannot be commuted. The noise driving should be constructed to be as statistically alike the command input signal as po ing the same power density spectrum and the same probability density function and higher orders). The parameters of c^ need to be adjusted to minimize the me-an square o The LMS algorithm cannot be used because error at the output of C is not avai can be done, however, by using linear random searchJLRS) or differential steep (DSD), described in Chapter 6. Once the weights of C are obtained, they can be c^ COPY to form the controller.
I
Noise
I I
b
+
COPY
2
F
-
I
I I I I
Reference model Offline process
Figure 11.7
A nonlinear model-reference adaptive inverse control system.
11.3.1 A Filtered-€ Technique for Nonlinear Adaptive In Control
The LRS and DSD algorithms are easy to apply and work well, but they are v converge and use much more data than LMS to achieve convergence with a giv misadjustment. Accordingly, there is good reason to attempt to develop a learn based on LMS to obtain C. It turns out that the filtered error algorithm can be this purpose. The adaptive inverse control techniques described in previous chapters, in filtered-X algorithm, cannot be used with nonlinear systems because they requir sible commutation of filtering operations. The ordering of filtering operations natural way with the filtered-c approach, and as long as signal levels and statist teristics are_maintainedappropriately, the filtered-c technique is applicable to t of finding C. The basic approach to the filtered-c algorithm is illustrated in Figs. 7.4 linear SISO control systems, and in Fig. 10.18for linear MIMO control systems. incorporates many of the features of these block diagrams. The operation of system is straightforward and needs no special description here. The system in
Command input
-
4
COPY
c^
I
/
Controller
lr
V Online COPY
F
Offline
-
Pi'
/
Noise generator
Overall
-
Reference model
M
Figure 11.8
The filtered-€ LMS system, suitable for linear control
The system of Fig. 11.9 is very similar to that of Fig. 11.8, except that w nor modification, it is now capable of doing nonlinear control. If dither is requi plant modeling pixess, schemes A N L ,B N L ,or CNLcould be used. The adapti works in the same way as that for finding (z) o for finding the nonlinear case. How well does the nonlinear inverse of the nonlinear plant function? Thi question. A lot of experience and theoretical development (not yet available) will
Fi'
Pi'
I
Command input
4
+
I
-
1
Cdntroller
Online Online
1
t
\
COPY
, L # +
F
Offline Noise generator
Delay A la l
I
+
Figure 11.9
Y
I
Reference model
M
A filtered-€ LMS system capable of nonlinear control.
It should be noted that when one is finding pi' offline, the noise genera have the same dynamic range and statistical character as the output of the cont ing the right dynamic range is critical to nonlinear inverse modeling, just as it modeling.
the Pi1COPY filters are identical to each other. An alternative to the system of Fig. 11.9 is shown in Fig. 11.10. @re the o is used in generatiEg the error signal for the adaptive process for finding C. The ad that the output of P does not contain plant disturbance. The dizdvantage is that d between P and P could bias the adaptive process for finding C. Thus, the contr differ from the ideal. The control engineer must trade advantages against disadv each practical case. Yet another alternative, this one an alternative to the system of Fig. 11.10 in Fig. 11.11. The plant input is used here instead of tJe plant output to obtain of the error signal f o r c e adaptive processior finding C. Instead of running the output through P or P and then through P i ' , one simply takes the plant inpu it directly as shown. A delay A must be included in the signal path. The beha system of Fig. 11.11 is similar to that of the system of Fig. 11.10. More work will tz determine which system is better and when. They both appear to work quite P;' is an accurate plant inverse.
11.3.2 Adaptive Inverse Control for Nonlinear MIMO Sy
The block diagram of Fig. 11.9represents a system capable of adaptive nonlinear SISO plants. It is able to be generalized, however, to represent an adaptive inve system for MIMO applications. To do MIMS, the form and structure of the syste same. Within the blocks, the plant model P would be structured like the adapt filter of Fig. 10.7, except that each 5omponent adaptive filter would be a nonline ~ apply to P i ' , used to filter the components of the over filter. The s a m remarks error, and to C, the controller. The reference model M would also be MIMO, could be nonlinear. An algebraic technique has been devised in order to demonstrate that the Fig. 11.9 will perform correctly for adaptive inverse control of nonlinear MIMO This simple form of algebra will be introduced here and it will prove to be hel analysis of nonlinear signal processing systems that are either SISO or MIMO. Refer once again to Fig. 11.9. We shall represent the input to the control I , This is meant to be a symbolic representation, not necessarily a z-transform o vector. If we neglect plant disturbance, we can represent the plant output as
z = PCII.
The notation is similar to that of finding the z-transform of $e plant output, as i were linear transfer functions. But in this use,_both P and C are nonlinear. Sym (1 1.1) represents that input I is first applied to C , whose output is then applied to
j Command input
Ideal controller
{
5 > .- . -
-
-
f
Controller
Online
Oftline
I
Reference model
M
Figure 11.10
An alternative filtered-r LMS system capable of nonlinear control.
;
Ideal controller
I
Command input
4
-
1
€'
*A. -
+
/
F
Online
I
! CO_pY
P
Offline
+
Noise generator
Delay A
Reference model
M
Figure 11.11
Another filtered-€ LMS system capable of nonlinear control.
Comparing the output of the actual controller with that of the ideal controller, th error 6' would be expressed as A
€I
= c*z - C I .
The error signal used in the adaptation of c^ is designated in Fig. 11.9by Z'. the block diagram and neglecting plant disturbance, this error signal can be exp = Pi'MI - P i ' P e l . A
Substituting (1 1.2) into (1 1.4) yields
El' = Pz;'PC*I - P i ' P F I .
The adaptive process for the formation of pi' is shown in Fig. 11.9ABy in is clear that if the error of this process is very small, then the c y a d e of P and equivalent to a delay A, given that the input to P^ is I through C or w h c is apE the same, I through C*. (It is interesting to note that the cascade of P and E equivalent to the delay A, but because of nonlinearity,the cascade of and P necessarily be equivalent to the simple delay A.) Refer back to (1 1S ) .Assume that F i s a close enough fit to P that Fcan be for P. Accordingly,
Pi'
Replacing pi'P^ by A yields
= P?;'FC*I -
pz;' PEI.
A
2' = AC*I - ACI = A(C*I - E l ) . In light of (11.3), we notice that
t' = A(€').
Accordingly, when adapting c^ to minimize the mean square of Z', we are min mean square of 6' and are thereby choosing a controller that is as close as pos ideal controller. In obtaining this result, wz have assumed that P^ closely fits is close to a perfect delayed inverse of P. It should be noted that Eqs. (1 1.1 not real equations. They are relations that help us keep track of the flow of sign systems operators and delays, whether linear or nonlinear, SISO or MIMO. Similar algebraic techniques can be used to show that the adaptive proc trated in Figs. 11.10 and 11.11 develop controllers that are close to the ideal f nonlinear plants, and for SISO or MIMO plants. On the other hand, the adaptive Fig. 11.8 is not necessarily ideal for nonlinear applications.
Pi'
= C*I - C I A
El
The question is: Are these two error signals the same? Regarding Fig. 11.8, we note that M I = PC*I. This is really a definition of C*. Substitution of (1 1.11) into (1 1.9) yields
( j;i)
Assume once again that
= P z ;' ( f C * I - PCII).
P^ is a very close fit to P, so that
Since P^ and pi' are nonlinear operators, we note that
F i ' ( F C * I - FEI) # A(C*I - ? I ) .
The reason is that for the left-hand side of (1 1.14), there are two signal CoInpone Pi'. If each component were acting separately,then the combination of f;' an be equivalent to A. The two components acting simultaneously cause this equi be invalid. Using (11.14), (1 1.13), and (1 1.lo), we may conclude that
Accordingly, adapting c^ in the system of Fig. 11.8 to minimize mean square err necessarily provide a controller that performs as closely as possible to the ideal c
NONLINEAR PLANT DISTURBANCE CANCELING
Disturbance can be canceled for both the SISO and MIMO cases, even when t nonlinear. Since MIMO is more general, a scheme for doing this with nonline plants is shown in Fig. 11.12. The methodology involved is based on the linear M noise-canceling scheme of Fig. 10.19 but adapted to work with nonlinear MIMO utilizing concepts developed for the systems of Figs. 11.4 and 11.9.
Controller output
Figure 11.12
\
[Ql
COPY I
z-’ I
Control system output vector
Nonlinear MIMO plant disturbance canceling with online formation of (FA)-’and Q.
the correct plant model and inverse derivative model PI must be available (see The input level for this derivative model is s$ by the actual plant input signal in Fig. 11.12. The inverse derivative model (PA)-’ is obtained by another onlin process, shown in Fig. 11.12. The driving function for the plant derivative model as that used in the subsystfm for generating Q, and the signal level is;et by the a input signal. Adapting ( P A ) - ] as shown provides the appropriate (PA)-’ for t for generating Q . An inspection of Fig. 11.12 allows one to verify that after convergence of and ( % ) - I , correct signal levels and signal statistics will be present at the input and (PA)-’in t k subsysem for generating Q , and the levels and-statistics will at the inputs of P’ and (PA)-’ in the subsystem for generating ( P A ) - ’ . The en will provide the capability for the optimal nonlinear plant disturbance zncelin means that for the chosen configuration of the filter Q and of the filters P’ and ( weights will be found to minimize mean square error and to minimize plant out bance power.
A N INTEGRATED NONLINEAR M I M O INVERSE CONT SYSTEM INCORPORATING P L A N T DISTURBANCE CANCELING
Figure 11.13 is a nonlinear MIMO system incorporating adaptiie inveLse dynam and plant disturbance csceling. Scheme CNLis used to obtain P and P’.Online which are used in the plant disturbance canceling for generating Q and (Pi)-’, identical to the corresionding processes in Fig. 1 1.12. The controller C is generated by an offline process driven by synthetic no the same dynamic range and statistics as the input command signal vector. Alt driving signal being the input command s this process could be online wig tor itself. In order to generate C, (PA)-’ is needed. A Focess to generate it is Fig. 11.13. This process is driven by the input signal to P in the suJbsystem for c^, providing the correct dynamic range and statistics for finding (PA)-’, Whengl the adaptive processes in Fig. 11.13 converge (finding P and p Q,FA, and C), the entire system will respond like M as best possible in the lea sense, with optimal canceling of the plant disturbance.
thz
Input command vector
I
(Online process for gene (Online process for generating [Ql)
#
4 . 14 I I
Delay
To provide background information for these studies, Appendix G contain of [6] by Widrow and Lehr which describes the backpropagationalgorithm used neural networks and explains how the LMS algorithm is at the foundation of bac tion. Appendix H shows alternative ways of structuring nonlinear adaptive filter tapped delay lines and neural networks and shows how adaptive inverse control c for MIMO plants using these structures and adapting with the filtered+ method In this section, we will use an adaptive filter of the type shown in Fig. 1 input signal is fed to a tapped delay line. The tap signals are used to drive the a linear combiner to make an adaptive linear filter. The same tap signals are a drive a two-layer neural network comprising a nonlinear adaptive filter. The out linear adaptive filter and the nonlinear adaptive filter are summed to make the ou The weights of the linear adaptive filter are adapted by LMS.The weights of the adaptive filter are adapted by backpropagation. The output could be compared t response to obtain an error signal. All of the weights are thus adapted by steep to minimize mean square error. This filter is like the one shown in Fig. 11.2. The filter of Fig. 11.4 can be used for nonlinear plant identification. The id trated in Fig. 11.15. The nonlinear dynamic plant was constructed for demonst poses by combining two linear one-pole digital filters in a cascade with a nonline The dynamic parts are linear. The nonlinearpart is memoryless. This plant was no to be a generic nonlinear dynamic system, but merely a good example of such a The adaptive system, containing linear and nonlinear parts, has the challen lem of modeling the plant and learning to imitate its behavior. The architecture o tive model is completely different from that of the plant. At the outset, there is no that values of the adaptive weights would exist to allow equivalent behavior for th the adaptive model, given a wide variety of plant input signals. The first experiment in plant modeling was done by setting and fixing th of the nonlinear adaptive filter to zero. The linear adaptive filter was adjusted to mean square error (refer to Fig. 11.15). After training, a sample of the nonlinear p was plotted simultaneously with the linear adaptive filter output. In Fig. 11.16 curve is the plant output, and the dashed curve is the adaptive filter output. The difference between the two curves and, not surprisingly,it is significant. One can to be able to model a nonlinear plant with a linear adaptive filter. The next experiment with the systems of Fig. 11.15 was done by adaptin linear and nonlinear filters simultaneously, making possible a nonlinear model o linear plant. After convergence, segments of plant output signal and model ou were plotted. In Fig. 11.17, the solid curve is the plant output and the dashed c nonlinear model output. The error between them is very small, indicating that th a good fit to the plant, at least for the given random input.
'
)
,
Figure 11.14
A linear/nonlinear adaptive filter.
4daptive near filter
/
h
I
/
Adaptive nonlinear filter
Figure 11.15
Nonlinear plant identification.
- Plant Output
I
____
-1
Adaptive Model Outpu
-3
I
-4 550
5'
I
Time (0.I sec.) Figure 11.16
Plant output and adaptive linear model output.
3 CI
.I
=Lo E d
-1
-2
- 550 -4
3
6w
650
Time (0.I sec.) Figure 11.17
Plant output and adaptive nonlinear model output.
When the adaptive model was only linear, the converged weights were training with the random input signal. The weights of the nonlinear filter were se sinusoidal test input signal was applied to the plant and the linear model and typ are plotted in Fig. 11.18. The solid curve is the plant output and the dashed c output of the linear model. The error between them is considerable, and this is Another experiment was done, using the converged weights of the nonli The linear and nonlinear filters were simultaneously trained with a random inp weights converged and were fixed. A sinusoidal test input signal was applied plant and the nonlinear converged model, and the outputs are plotted in Fig. solid curve is the plant output and the dashed curve is the output of the nonlin The curves do not match perfectly, but they are very close. These experiments were done by Bradley Smith. They are interesting, an gest that it will be possible to make good direct and inverse models for nonline plants. But there is no doubt that much work remains to be done before this kind o becomes routine.
11.7
SUMMARY
This chapter develops a set of fundamental techniques for using adaptive inve with nonlinear plants, both SISO and MIMO. The chapter begins with block d nonlinear tapped delay-line filters based on Volterra series and on neural netw weights of the Volterra filter are adapted by the LMS algorithm. The weights
Model Previously Trained with Random Signal -4
0
I
100
I
200
I
3W
I
400
I
500
I
600
Time (0.I sec.)
I
I
I
700
BM)
900
Figure 11.18 Plant output and linear model output with sinusoidal test input signal. Linear mod and fixed after training on random input.
____ Plant Output
Sinusoidal Test Input
-3
Adaptive Model Output
Model Previously Trained with Random Signal I
I
I
I
I
I
I
I
I
Figure 11.19 Plant output and nonlinear model output with sinusoidal test input signal. Nonlinea verged and fixed after training on random input.
ically small in amplitude compared with the plant input signal, would excite o signal response on top of the large plant input response. Proper modeling of th be done by making use of a derivative model rather than an actual model. The the derivative model are in turn used to obtain the weights of the actual model derivative model idea, nonlinear versions of schemes B and C are developed. Finding the inverse of a nonlinear plant is described next. Although nonl do not have exact inverses, it is possible to determine weights for a nonlinear f for a given input signal, the cascade of the nonlinear filter and the nonlinear pla duce an output that is a best least squares match to the output of a reference m by the same input signal. The weights of the nonlinear filter were adapted by th LMS algorithm. Methods for plant disturbance canceling are developed next. With small icant differences, they are quite similar to those used with linear plants. Both online techniques may be used to find Q. Having all the parts, an entire nonlinear adaptive inverse control system is featuring separate control of plant disturbance and separate control of nonline namics. MIMO as well as SISO designs are presented in this chapter. Since nonli tors are not commutable and MIMO systems components are also not commutab the systems MIMO did not significantly add to the conceptual complexity. At lea diagrams for nonlinear MIMO systems are not much different from those of non systems. In working with nonlinear plants, care was exercised not to commute no erators. The same was true for MIMO operators, even if they were linear. An a developed for nonlinear and MIMO signal flow to help in the design and analy systems. Whenever one is adapting a nonlinear filter, SISO or MIMO, the input s and statistical characteristics must be the same as will be encountered in actual the final system. We believe that the proposed methods for finding the weights of a nonlin controller and the weights of an adaptive plant disturbance canceler are optim to the best least squares solutions possible for the chosen configurations of adap Much work needs to be done with nonlinear techniques for adaptive control. S questions are beginning to emerge. Surprisingly, even some of the answers are to emerge as well. There are great opportunities here.
[3] P. WERBOS,“Beyond regression: New tools for prediction and analysis in ioral sciences,” Ph.D. diss., Harvard University, August 1974.
[4] D.B. PARKER, “Learning logic,” Tech. Rep. TR-47, Center for Comput search in Economics and Management Science, M.I.T., April 1985.
[5] D.E. RUMELHART, and J.L.MCCLELLAND, Parallel distributedprocess and I1 (Cambridge, MA: M.I.T. Press, 1986).
[6] B. WIDROW,and M.A. LEHR,“30 Years of adaptive neural network tron, Madaline and backpropagation,” Pmc. IEEE Vol. 78, No. 9 (Septem pp. 1415-1441. Neural network learning (Cambridge, MA: M.I.T. Press, [7] S.I. GALLANT, Neural networks (New York: Macmillan, 1994). IS] S. HAYKIN,
[9] R. HECHT-NIELSEN, Neurocomputing (Reading, MA: Addison-Wesley, 1
[lo] J.A. HERTZ,A. KROGH,and R.G. PALMER, Introduction to the theory computation (Reading, MA: Addison-Wesley, 1991).
[ 111 B. KOSKO,Neural networks and fuzzy systems (Englewood Cliffs, NJ: Pre
1992).
[ 121 P. MEHRA, and B. W. WAH,Ed., Artijkial neural networks: Concepts and t
Alamitos, CA: IEEE Computer Society Press, 1992).
[ 131 P.K. SIMPSON, Artificial neural systems (New York: Pergamon Press, 199
[14] M. KAWATO,Y. UNO, M. ISOBE,and R. SUZUKI, “Hierarchical neura model for voluntary movement with application to robotics,” IEEE Contr Magazine, Vol. 8, No. 2 (April 1988), pp. 8-16.
[15] D. PSALTIS,A. SIDERIS,and A.A. YAMAMURA, “A multilayered neur controller,” IEEE Control Systems Magazine, Vol. 8, No. 2 (April 1988), p
[I61 W.T. MILLER,R.S. SUTTON,and P. Werbos, Eds., Neural Networks f o (Cambridge, MA: M.I.T. Press, 1990).
While developing the principal mathematical results of adaptive inverse control, we come upon a result from time to time that seemed to us to be a pleasant surprise. of these results were desired and were anticipated intuitively. Some were easy to pro some required a great deal of algebra. We were hoping that the desired results would because if they were, it would make the theory simple and the applications easy. Wh one of the desired results was proven to be true, we said to ourselves: ‘This is am There must be something right about this approach to adaptive control.” The purp this brief chapter is to review the pleasant surprises and to summarize the findings book.
1. Precise inverse controllers can be devised for minimum-phase plants and, with
what delayed response, for nonminimum-phase plants too.
2. The effect of closed-loop response can be obtained in an open-loop feedforwa trol system by using the feedback inherent in adaptive filtering to find the i adaptive controller.
3. Plant disturbance can be optimally canceled using feedback with zero gain the loop. Best linear least squares plant disturbance canceling can be accomp without altering the plant transfer function.
4. Plant disturbance canceling can be done independently of plant dynamic contro
optimization of one of these processes is not compromised by the optimization other.
5. Achieving an overall system response which is a best least squares estimate of a ence model’s response is generally straightforward and natural with adaptive i control.
6. If the plant is unstable, it must first be stabilized by feedback. Then the plant feedback are subject to adaptive inverse control, treating the plant and its stab feedback as an equivalent plant. The ability to cancel plant disturbance is una by the choice of stabilizing feedback. The ability to achieve a desired overall dynamic response is also unaffected by the choice of the stabilizing feedback inverse needs delay for its realization, the required delay will not depend on the of the stabilization feedback.
330
back of the plant disturbance canceler. Thus, the interaction of the multip processes in the integrated system cause its overall dynamic response to h an equilibrium condition which is a best least squares match to the refere response. The adaptive feedback provides robust behavior for the overall 9. The use of adaptive feedback does not create stability problems, except du up or during a sudden catastrophic change in plant dynamics when the pl bance canceling loop could get out of balance and go unstable. The remedy porarily abstain from plant disturbance canceling by breaking the disturba ing loop until the plant model regains a response close to that of the plant 10. Adaptive inverse control applies readily to the control of MIMO systems time in a MIMO system goes up only linearly with the number of chann than with the square of the number of channels.
11. Adaptive inverse control applies readily to the control of nonlinear system SISO or MIMO. 12. Dynamic control of either a minimum-phaseplant or a nonminimum-phas certainly be accomplished with adaptive inverse control. But what of a plan zero exactly on the unit circle in the z-plane? The inverse of such a plant w to have a pole on the unit circle, and it would be unstable with either a l or a right-handed impulse response. Here is a case where inverse control s But it does not seem to, and this is surprising. We finish this chapter wi experiments making inverse controllers for plants with zeros which are ve unit circle and in some cases, exactly on it.
Figure 12.1 shows the impulse response of a stable, discrete, disturbance having two poles and one zero. The poles are complex conjugates which corre damped oscillatory response. The transfer function of the plant is given by ( Z - 1.05) P(z) = [Z - (0.9 0.3j)][z - (0.9 - 0.3j)l'
+
The plant is nonminimum-phase since its zero lies close to but just outside the u The impulse response of a delayed inverse filter is shown in Fig. 12.2. The delay w arbitrarily to be 100 sample periods, although it was really not necessary to have s delay in order to get an excellent inverse. Figure 12.3 shows the convolution of the impulse response of the plant a layed inverse. It is almost a perfect delayed impulse. Figure 12.4 is a plot of the put when driven by the inverse filter as a controller. The command input was w Superimposed is a plot of the output of a pure delay of 100 sample times. It is dri
-0.d 0
Figure 12.1
20
40
100
80
60
120
140
160
Time (0.1 sec.)
Impulse response of a discrete damped oscillatory nonminimum-phase pl
1
-0.4
1
0
20
Figure 12.2
40
60
80
100
120
140
160
180
Time (0.1 sec.) Delayed inverse impulse response of nonminimum-phase plant.
-0.21 0
20
40
60
80
100
120
140
180
180
Time (0.1 sec.) Figure 12.3
Convolution of impulse response of nonminimum-phase plant and its delayed invers
2.5
2 1.5 1
-
B a
.--
*z
0.5
0 -0.5
-1 -1.5 -2 -2.5
0
5
10
15
20
25
30
35
40
45
Time (0.1 sec.) Figure 12.4
Actual plant output and ideal output for nonminimum-phase plant.
Figure 12.5
Impulse response of a discrete damped oscillatory minimum-phase plan
The convolution of the impulse responses of the plant and its delayed shown in Fig. 12.7. The output of the plant driven by the inverse filter as a co the output of a delay unit of 100 sample periods are plotted in Fig. 12.8, both dr same command input. The two plots superimpose with almost perfect registrat there is no surprise here. Figure 12.9 shows the impulse response of another plant, similar to the pr plants. In this case, the poles are the same but the zero is exactly on the unit transfer function is (2 - 1) P (2) = [ z - (0.9 0.3j)][z- (0.9- 0.3 j ) ] '
+
This plant is not minimum-phase or nonminimum-phase. Its inverse transfer fu tains the factor 1/(1 - z - ' ) . Either to the right or to the left in time, the corresp pulse response could not be approximated by an FIR impulse response. If we w an IIR inverse, it would not be stable. We should not be able to construct an the plant represented by (12.3). But we did it! Figure 12.10 shows the impul of a delayed FIR inverse, with the delay set to 100 sample periods. Figure 12.1
"i -0.8 -1
0
20
Figulp 12.6
40
80
80
100
1x1
140
180
180
Time (0.1 sec.)
Delayed inverse impulse response of minimum-phase plant.
0.9 0.8 0.7 Q)
0
0.8-
.-
0.5
-
4
0.4
-
i ?
E
0.3-
0.20.1 -
01 0
Figure 12.7
20
40
80
80
100
120
140
180
180
Time (0.1 sec.)
Convolution of impulse response of minimum-phase plant and its delayed inverse
-1'51 -2
-2.5
0
5
10
15
20
25
Time (0.1 sec.) Figure 12.8
30
35
40
45
Actual plant output and ideal output for minimum-phase plant.
convolution of the impulse responses of the plant and its delayed inverse. This delayed impulse, not far from perfect. Using the delayed inverse filter as a contro the plant, the plant output is plotted in Fig. 12.12. Superimposed is a plot of th a delay of 100 sample periods, driven by the same command input as the contr plant. The two plots are very close, although they do not register perfectly. These experiments exhibit a continuum of behavior as the plant zero is m outside the unit circle to the unit circle itself, to inside the unit circle. Inverse co easily whether the plant is nonminimum-phaseor minimum-phase,with the zero close to the unit circle or far from it. It also appears to work reasonably well even exactly on the unit circle. These are surprising results. We don't know why this not clear that this will work with all possible zero placements on the unit circle. edge of inverse control of dynamic systems is based on the theory of two-sid transforms. There may be some other way to analyze and explain these things not yet understand.
-0.2
-
-0.4
-
-0.6
-
-0Bh 0
50
100
200
150
Time (0.1 sec.) Figure 12.9
Impulse response of a discrete damped oscillatory plant with a zero exactly on the
-0.4
-
-0.8
-
-0.8
-
-1 IJ
0
Figure 12.10
20
40
80
80
100
120
140
180
180
Time (0.1 sec.) Delayed inverse impulse response of plant with zero on the unit circle.
-0.2
Figure 12.11
0
20
40
60
80
1W
120
140
180
180
Time (0.1 sec.)
Convolution of impulse responses of plant with zero on unit circle and its delaye
troller.
2.5 2 1.5 1
4
0.5
-
0
.-3
*E
-0.5 -1 -1.5
-2 -2.5
0
5
10
15
20
25
30
Time (0.1 sec.) Figure 12.12
35
40
45
Actual plant output and ideal output for plant with zero on unit circle
Stability and Misadjustme of the LMSAdaptive Filt
In this appendix, we will analyze stability and misadjustment of the adaptive trans filter whose weights are controlled by the Widrow-Hoff LMS algorithm. The adaptiv is comprised of a tapped delay line, shown in Fig. 3.1, whose tap signals are applied adaptive linear combiner in order to generate the filter output.
A.l
T I M E CONSTANTS A N D STABILITY OF T H E M E A N OF T H E W E I G H T VECTOR
At each moment of time k, the input vector to the weights is Xk defined by (3.1), th output is yk defined by (3.3), the weight vector is Wk defined by (3.2), and the erro defined by (3.4). The Wiener solution is W*, defined by (3.10), and the difference be the weight vector Wk and the Wiener solution is defined as Vk and is given by (3.14 We will analyze stability of the LMS adaptive filter by examining it in the appli of modeling an unknown system in the configuration of Fig. A. 1. A common input xk is applied to both the adaptive filter and the fixed unknown filter. The adaptive f modeling the fixed filter. The fixed filter is assumed to have an independent internal which is represented by an additive zero-mean noise nk at the filter output. The diff between the noisy output and the adaptive filter output is the error signal E& which i by the LMS algorithm to adapt Wk. We assume that the adaptive filter has enough w to perfectly match the dynamic behavior of the fixed filter. Therefore, the impulse res of the fixed filter is the Wiener solution for the adaptive filter. If the adaptive weight set to their Wiener values, the error ek would be equal to the noise nk. The minimum square error of the adaptive filter is accordingly, tmin
=~ [ n i ~ .
A study of stability of the LMS algorithm as used in the context of Fig. A. 1 will perfectly general, but it will be applicable to a wide variety of practical cases. The res analysis leads to a simple stability criterion that works not only for systems configu like in Fig. A.l, but it is known to work for many other configurations not covered following derivation [2].
Xk
wk
Adaptive filter
Figure A.1
Adaptive modeling configuration used in study of LMS stability.
Inspection of Fig. A. 1 shows that the error € k can be expressed as Ek
= ( w ' -Wk)'xk
-vlxk + nk + nk.
+nk
= = -x:vk
The LMS adaptation rule (3.35) requires that
wk+I = wk + 2pEkXk. Subtracting W* from both sides yields Vk+l = vk
+ 2pCkXk.
Substituting (A.2) into this expression gives
+ +
+
Vk+l = vk 2 F ( - x l v k n k ) X k , Of Vk+l = vk 2 p X k ( - X l V k + n k ) , or Vk+l = (I - 2Fxkxi)vk 2@nkX&.
+
Now, we take expectation of both sides to get
E [ V k + l I = E [(I - 2pXkXl)Vk] *
The last term of (A.4) vanished under expectation since the independent noise related with input x k . Assume that is small. With slow adaptation, the adaptive weight vec on many past input samples, and its dependence on the present input vector i Accordingly, we assume that the input vector xk is independentof the weight ve therefore it is independentof vk. This same assumption was first made in [ 11. It
E[Vk+ll = E[Vkl -2PE[XkX:]E[Vkl.
E[V;+,] = (I - 2/A)E[V;]. This equation is in pure diagonal form, and its solution is
,
E[V;+,l= (1- ~ P . . ~ ) ~ E [ V ~ I ,
where Vb is the initial value of Vk expressed in the rotated (primed) coordinate premultiply both sides by Q to get E[Vk+i 1 = Q(I - 2pNkQ-'
W"l.
As k goes to infinity, the expected value of Vk+l will go to zero as long as all the ratios of (A. 11) are less than unity, that is, Jrp< ( 1, for all p, where r p = (1 - 2wAP).
Therefore, the condition for stability of the mean of the weight vector is 1 >p
> 0, or
Amax
1 > PAmaX > 0.
This is in accordance with Eq.(3.27). Since the geometric ratios given by (A. 12 tical to (3.28), it follows from (3.30) that the pth time constant is
For the special case of the input being white, all eigenvalues of R could be
Eq. (A. 13) becomes
n
->p>O,
tr R
1
E[x:I-O* The single time constant for this case is
or
when operating with LMS in the vicinity of the Wiener solution. Accordingly,
[
weight noise coLa$,
]
= E[vkv:] = CI.
The variance of the weight noise, a scalar, can be defined as
From (A.16), uk
= nc, and
1 E[vkv:] = - V k I . n To establish stability of the variance of the weight noise, we shall examine the v& during LMS adaptation. Refer to Eq. (A.4). hemultiply both sides of this e the respective transposes to obtain
v:+, Vk+l = v:(I
- 2kxkXf)2Vk + 4w2niX:Xk
+4knkX: (I - 2&x:)vk.
Next, we take expectation of both sides. The last term of (A.20) disappears und tion because the noise nk has zero-mean and is independent of xk and vk. The of the next to last term of (A.20) is factorable because of these same properties Thus E[VkT,lVk+l]= E [ v r ( I - 2hxkxl)2vk1
+4w2E[ni] tr R. For convenience in the utilization of (A.22), we define a scalar a as U
A
= E [Vl(I - 2@xkx:)2vk].
= tr ( E [v~v:(I- 2 p ~ ~ x ; ) ~ I ) .
Recall that p was chosen to be small (corresponding to slow adaptation), so the w tor Wk and the weight deviation from the Wiener solution Vk are essentially in of the input signal vector Xk. As such (A.25) can be expressed as a = tra = tr ( E [ V k V [ ] E[(I - 2pXkX;)2]).
Now,
E [(I - 2 / ~ X k X l ) ~=] E [I - 4pXkX:
+ 4p2Xk(X,TXk)Xl].
The scalar quantity (X:Xk) does fluctuate as the input vector varies. If we assum that the number of weights is large (i.e., let n be large), the law of large number that this scalar quantity, the magnitude square of the input vector, will be essen stant. Accordingly,
(XlXk) = n E [ x f ] = tr R. Equation (A.27) can now be written as
+ 4p2 tr R(XkX,T)]
E [(I - ~ / . . L X ~ X : )=~ ]E [I - 4pXkX: = I - 4pR
+ 4p2R tr R.
We may now substitute (A.29) and (A.19) into (A.26) to obtain [I - 4 p R
+ 4p2R tr R]
1 = -uk tr (I - 4 p R
+ 4p2R tr R)
1 = -vk (n - 4p tr R n
+ 4p2(trR)2).
n
Next we substitute (A.30) into (A.23), and substitute the result into (A.22) to ob
n +4p2 E [nil tr R.
n
This geometric ratio is a quadratic function of p that can be shown to never g Equation (A.33) is therefore equivalent to 1-4P t r R + -(trR)2 4w2 n n
< 1.
This is equivalent to R)' < 4w -tr R.
*(tr n
n
Since trace R is positive and since the number of weights n is positive, p must to satisfy (A.35). Dividing through both sides of the inequality by the positive 2 t r R, n
we obtain the stability condition O
Satisfaction of (A.36) assures stability of the variance of the weight vector. Since stringent condition than (A.13), satisfaction of (A.36) assures both stability of t of the weight vector and convergenceof its mean to the Wiener solution. Assum in the derivation include small p, large n , a scalar weight noise covariance matri tation for the purpose of plant modeling (in the context of Fig. A. 1). Further use of Eq. (A.3 1) yields a more refined derivation of misadjustm one obtained in Chapter 3. Having chosen p within the stable range of (A.36) a enough adaptation time for Eq.(A.31) to come to equilibrium, it is clear that Uk+l = Vk. Substituting (A.37) into (A.31) gives the variance of the weight noise as uk
= n P m 3 1 - ptr R '
The weight noise covariance matrix can be obtained from this, using Eq. (A.19 weight noise covariance matrix
1 1 - /L tr R '
Approximations were made in the derivation of (3.60) that were not made in the of (A.40), such as the neglect of terms dependent on p2 that in a small way contr misadjustment. The original formula for misadjustment based on the noise covarian (3.60) is M=pCLR.
Using the new expression for weight noise covariance provides a new formula justment that is derived simply by incorporating the factor (A.41). Accordingly
M =
p trR
1- p t r R '
This new refined formula predicts misadjustment more accurately. For small val is identical to the original formula. Experimental verification of (A.43) has been done in comparison with (A.4 eling experiment was done following the configuration of Fig. A. 1. Results are g ble A. 1 for an experiment with an adaptive filter containing 20 weights. The in to both the adaptive filter and the fixed filter was white noise of unit power. Th trace R was 20. Both uniformly distributed and Gaussian signals were used, an results were obtained. The new more refined formula was a more accurate predic adjustment, especially for large values of p. For misadjustment levels below there was not much difference, but for larger levels of misadjustment, the new fo much more accurate. Refer to Table A. 1. Horwitz and Senne [2] have presented a derivationof stability of the varianc obtained stable bounds on p that are somewhat more precise, but very similar Their earlier derivation is more complicated. However, it is based on assumption that we have used. They also present a refined misadjustment formula that is same as (A.43), but once again their derivation is more complicated. Their paper h been of very high interest to researchers in the field. It should be noted that there are more general proofs of convergence of th gorithm (see [3] and [4]), which do not necessitate all the assumptions that we h above. On the other hand, these proofs are valid for very small values of adaptatio p (i.e., for values of p much smaller than the practical stability range (A.36)).
0.015 0.02 0.025 0.03 0.035 0.04 0.045 (*):
A.3
0.438 0.676 1.050 1.640 3.00 6.23 (*)
0.30 0.40 0.50 0.60 0.70 0.80 0.90
0.429 0.667 1 .00 1S O 2.333 4.00 9.00
sometimes blows up
A SIMPLIFIED HEURISTIC DERIVATION OF MISA D J UST MENT AN D STAB I LI T Y C0NDITI0NS
The same results obtained in A.2 can be obtained in a much simpler and more int ner. Assume that the LMS adaptive process has converged so that the mean of vector is equal to W*. The minimum mean square error is thin, and for the mo figuration of Fig. A.l, this is equal to the plant noise power E[n:]. Because of n weights, the actual mean square error will be greater than emin.The weights wi going random brownian motion about the Wiener solution at the bottom of the Excess mean square error is created by the input signal propagating throug weights. The average mean square error can be expressed as
t = tmin +
( FF: ) average
The idea of average excess mean square error has been discussed in Chapter 3. The noise in the weights is uncorrelated from weight to weight. With an ad of the FIR type, the average excess MSE will be equal to the input signal power by the sum of the variances of the weight noises. Accordingly, variance =E[x;].n. ofnoiseof ) . single weight
(E :)
(
The covariance matrix of the weight noise is derived in Eq. (3.60) and is cov W k l = &ninI.
The weight noise power is proportional to &,in, according to this formula. Thi approximation when operating near the bottom of the bowl, adapting with a sm p. Otherwise an appropriate heuristic formula is the following: cov [Vk]= pg1.
This is very similar to Eq. (3.64) with 6 used in the expression in place of .$,in. mean square error can be obtained from (A.44) and (A.48):
‘= -
T
tmin
1-p trR’
The misadjzstment is defined by (3.65). Combining with (A.49), we have
&- t m i n tmin
-
p tr R
1-ptrR’
This result is the same as that of (A.43). Stability can be determined by keeping M, given by (A.43) or (A.50), fini tion, we know that p must be positive for stable adaptation. Accordingly, to ha adaptive process, we must choose p from within the range 1 ->p,>O, or tr R
This is the same as (A.36), the criterion for stability of the variance of the LMS
Bibliography for Appendix A
[ l ] B. WIDROW,“Adaptive filters,” in Aspects of Network and Syste ed. R.E. Kalman and N. De Claris (New York: Holt, Rinehart and Wins pp. 563-587.
[2] L.L. HORWITZ, and K.D. SENNE, “Performance advantage of complex LM trolling narrow-band adaptive arrays,” ZEEE Trans. on Circuits and Systems 28, No. 6 (June 1981),pp. 562-576.
Comparative Analys of Dither Modelin Schemes A, 9, and
Chapter 4 describes three plant modeling schemes involving the use of a random dith nal. The purpose of this appendix is to analyze these schemes and to compare their p mances so that their relative advantages and disadvantages can be delineated. We are concerned with convergence conditions, speed of adaptation, misadjus and the noise in plant estimation (the weight noise covariance matrix). For each sc faster adaptation causes more noise in the weights of the plant model. The schemes compared by evaluating their weight noise covariances for a given speed of adaptati Before doing these analyses, it is convenient to make certain definitions. Refem Figs. 4.3(a), (b), and (c), we will denote the last n samples of the controller output a
u; A (u;, Ui-1, . . . u;-,+j Y,
the last n samples of the dither as
A Ak = (6k9 6 k - 1 ,
-
*
*
9
ak-n+l)
9
Uk-n+l)
T
7
and the last n samples of the plant input as
uk =A (ukq uk-17
9
.
.
T
.
For these signals, we may define autocorrelation matrices and corresponding values as A
Rut = E[U;Uir], with eigenvalues
R A = E[AkA:], A
with eigenvalues
plant. (If instead we were to assume that the plant model has more taps and w the plant, the following analysis would be unchanged. That would not be the plant model has fewer taps and weights than the plant.)
8.1 ANALYSIS OF SCHEME A
The configuration of scheme A, shown in Fig. 4.3(a), is similar to Fig. A. 1 exc plant input and plant model input with scheme A consist of the sum of the dithe the controller output signal. We can make direct use of the analytical results o A. From (A. 13a), the pth time constant for scheme A is 1 tp = -. 2cLApu The plant input autocorrelation matrix is Ru = Rut + RA. If the dither is white, the eigenvalues add, so that ApU = k p U 1
+
ApA.
Since all the eigenvalues of the white dither are equal, APa = ha = E[Si].
The time constant can be expressed as
The condition for stability of the variance of the weight vector with scheme A tained from (A.36) as
It may be noted that t r R ~= t r R u e + t r R ~ = n(E[u;.2]E[S,2]).
+
The stability of the variance is now conditioned by
The misadjustment of the adaptive model of scheme A is obtained from (A.43)
M =
8.2
CLwRu 1 - p tr RU
-
+ ErS;I) 1 - pn (E[uL2]+ Era;]) (Eb3
’
ANALYSIS OF SCHEME B
Scheme B is shown in Fig. 4.3(b). To facilitate the analysis of this system it is to redraw the diagram in the form shown in Fig. B.1. The resulting adaptive m will be the same. In Fig. B. 1, the effect of the control signal u; driving the plan a disturbing influence from the point of view of the modeling process, adding t disturbance nk to give an equivalent plant disturbance i i k .
Controller output
y;
4
Plant
,-+ Error €k
I Figure B.l
I
Adaptive model
F(Z,
I
Alternate representation of plant modeling scheme B.
The configuration of Fig. B.l is very similar to that of Fig. A.l, so that th of Appendix A can be directly applied to the analysis of scheme B. Assuming w the time constant of scheme B is obtained from (A.13a) as
weight noise covariance matrix If the response of the plant to the controller output u; is represented by equivalent plant noise is f i k = y; nk.
+
Since y; and nk are mutually independent,
Ern;] = E[y;*] Equation (B.23) can now be rewritten as weight noise covariance matrix
]
+ Ern;].
= P (E[YLzl-4-
Em:])
1 - pnE[8,2]
I.
The misadjustment for scheme B can be obtained from (A.43) as
M =
PtrRA - Pn E[8,21 1 - LL tr RA 1 - pnE[8,2]’
6.3 ANALYSIS OF SCHEME C
Scheme C is a more sophisticated adaptive process than either scheme A or sch not in any way similar in configuration to the system of Fig. A. 1. The analysis o A, although helpful in methodology, cannot be used directly to analyze scheme “start from scratch.” Refer to Fig. 4.3(c). Let P and $k represent the plant impulse response ve kth estimation oLthe plant impulse response vector correspondingly. We assu Wiener solution P* is equal to P, (assuming as was done above that the impul Pk is long enough to match that of P). APk will denote the weight error vector
A
A
A
A
AP = Pk - P* = Pk - P.
Using this notation, we can represent the adaptive modeling error E k as Ek
+
+
= nk (Ak u ; ) T p- (Ak = nk - (Ak + u ; ) T A P k .
+ u;)T$k
The LMS adaptation rule can be written in accord with (3.35) as A
A
Pk+l = pk
+ 2pEkAk.
E[APk+i] = E[I- 2/1.&(& +u;)T]E[APk] = (I - 2/1.E[AkAl]). E[APk]. Since the dither is white,
E[APk+ll = ( 1 - 2/1.E[$l)E[APkl.
Clearly the mean of APk converges to zero if and only if the geometric ratio has a less than one, that is,
The time constant of this convergence process will be
in accordance with Eq. (3.31). The convergence in the mean does not necessitate the convergence of the v the weights. In order to analyze the convergence of the variance, we have tozva the covariance of the weight noise. According to_Eq. (A. 16), when the model Pk approaches the vicinity of the Wiener solution P",the weight noise covariance comes scalar. weight noise covariance = E[APk. APT] = CI. matrix
(
)
The question is, once the adaptive process arrives at the Wiener solutio weight vector remain in the vicinity of this optimal solution? In order to answer th we will evaluate the impact of each iteration on the variance of the weight noise, Vk
A E[APlAPk].
According to (B.35) Vk
= nc, and
Substitution of (B.371, (B.41), and (B.44) into (B.43) yields uk
(I - 4pE[S:]I -k 4p2nE[8:] . E[(& n Since U; and Ak are independent of each other, a = -tr
Vk
a = - (n - 4pnE[8:] n
+ u;)(Ak + u;)']).
+ 4p2n2E[8i](E[8,2]+ E [ ( u ; ) ~ ] ). )
Substitution of (B.46) into (B.40) yields uk+l
= u k { 1 - ~ F E [ ~ , Z-I [pn(E[8,21+ ~ E[(u;)2])1)
+4& E [nilE [8 3 .
According to (B.38) and (B.47) the variance of the adaptive process conve only if
+
1 - 4pE[S,21[1 - ~ ( E [ 8 , 2 1 E [ ( u ; ) ~ ] )<] 1.
Since E[8,21is strictly positive and E [ ( u ; ) ~is] nonnegative, (B.48) is equivalen
+
1 - pn ( ~ [ 6 , 2 1 ~ [ ( u ; ) ~>] 0, ) with p > 0.
This stability condition has been verified for dither scheme C by extensive sim periments. Use of Eq. (B.47) allows an accurate but simple derivation of misadjust made. Choosing p in the stable range (B.50) and allowing time for the adaptive come to equilibrium, we have Vkfl
= uk*
Substituting (B.51) into (B.47) gives the variance of the weight noise as vk =
PnE[n;I 1 - p n (ELL$+ ~ [ u f '] )
The weight noise covariance matrix can be found from this, in accord with (B.3 weight noise covariance matrix
]
= kukI =
PE[41 I. 1 - p n ( ~ [ 6 , 2 1~+ [ u f ] )
For scheme C (Fig. 4.3(c)), the minimum mean square error is the plant disturba Therefore, (B.53) can be rewritten as
Excess mean square error in the modeling process will be caused by we The modeling error is given by (B.29). This includes the effects of plant disturb error caused by weight noise, apart from the effects of plant disturbance, is give error due to weight noise Since
I
= -(&
+U;)TAPk.
uk = Ak +u;, we can write this as error due to weight noise The mean square of this error is
I
= -ufAPk.
E[U[U,] = tr R u , and since 8k and u; are independent of each other, n (E[8:]
+ E[uL2])= tr Ru.
Substituting (B.60)and (B.61)into (B.59)gives
The misadjustment is obtained by normalizing this with respect to the minimum cordingly,
A SIMPLIFIED HEURISTIC DERIVATION OF MISADJUSTMENT AND STABILITY CONDITIONS FOR SCHEME C
It is possible to use the analytical techniques developed in Section A.3, to prod plified analysis of stability and misadjustment for the adaptive process of schem Refer to Fig. 4.3(c) for a block diagram of scheme C. The minimum mean s is, as stated above, equal to the power of the plant disturbance. The average m error of the converged adaptive filter (when the mean of the weight vector equals weight vector) is equal to the minimum mean square error plus the average ex square error due to weight noise:
average
The weight noise of P^(z) is, of course, identical to the weight noise of P^(z) co these filters contribute to the_average excess mean square error. The adaptive model P ( z ) , which has noisy weights, contributes to aver mean square error in the following way. Since the noise in the weights is uncorr weight to weight, the average excess mean square error component at the outp will be equal the input dither power multiplied by the sum of the variances of noises. Therefore 1 varianceof 1 -8k21' ;MSE ' single weight
(
The covariance matrix of the weight noise is COV[Vk] = p61.
The reasoning is already given in Appendix A in connection with Eq.(A.47). Th of the noise of a single weight is therefore equal to p i . Substituting this into (B
This can also be expressed as
Combining (B.70) with (B.64) gives
The misadjustment can be calculated as follows,
M=
(average excess MSE) lmin
- np{E[u:] lmin
*
Substituting (B.71) into this expression gives
This result is identical to (B.63). Stability requires that p be positive and small that the misadjustment remains finite. Accordingly,
1 ->>>0. tr RLI
This is the same as criterion (B.50) for stability of the variance when using schem simple heuristic derivations of misadjustment and criteria for stability for sche yielded results that are identical to those that have been obtained with more rigo ment in Section B.3.
The adaptive plant model P^(z) was FIR. The number of weights of the plant chosen to be n = 10. The system of Fig. 4.3(c) has three input signals: the controller output u 8 k , and the plant noise nk. For the sake of simplicity, we have chosen all three zero-mean, white, of unit power, and independent of each other. Keeping all of the above design parameters fixed, we varied the adaptation For every choice of p , we have run the system-of Fig. 4.3(c) for a few thousand d starting each time with the initial condition Pk(z) =,O fork = 0. We monitored squares of the differences between the weights of P ( z ) and the weights of the P ( z ) . Learning curves were obtained based on the plant model error as a funct From these curves, we have computed the experimental time constant t. Equ gives a theoretical time constant formula for scheme C. For various values of p and experimental values of time constant were taken, and the results are sho B. 1. Discrepancy between theoretical and experimental time constants is of 10 percent or less, over a very wide range of time constant values.
TABLE B.l COMPARISON OF EXPERIMENTAL AND THEORETICAL VALUES OF TIME CONSTANT AND WEIGHT NOISE FOR PLANT MODELIN WITH SCHEME C Uk, Variance of
Adaptation constant
Time constant r Estimated Measured
the noise in
the weights of plant model Estimated Measured
~~
0.00025 0.00100
0.00200 0.00400 0.01 0.02 0.03 0.05
2000 500 250 125 50 25 17
2028 567 278 I28 53 20 16
0.0025 0.0102 0.0208 0.0435 0.1250 0.3333 0.7500 unstable
0.0024 0.0096 0.0223 0.0433 0.1331 0.3275 0.8302 655
Next we allowed the adaptive process to run for at least four time consta statistical equilibrium. In statistical steady state, the average noise in the weights model was measured. For each value of p, a considerable amount of avzagin to get accurate experimental values of the variance of the weight noise of P ( z ) . values of this parameter were obtained by using Eq. (B.52). A comparison is sho
By and large, Eq.(B.53) appeared to work quite well, but not perfectly. TABLE 6.2 WEIGHT NOISE AUTOCORRELATIONMATRIX FOR PLANT MODELING WITH SCHEME C 0 - 1 0 0 - 1 0 1 - 1 0 0 - 5 1 1 - 1 - 2 I 9 - 1 0 - 2 0 0 2 9 2 0 I 0 0 -I 1 - 5 - 1 0 -1 I 0 2 12 - 1 0 -2 1 -1 0 1 - 2 0 - 1 7 0 2 0 0 1 - 1 0 I 0 0 8 1 0 1 7 - 1 0 - 1 - 2 0 0 - 2 2 0 I 0 0 - 1 9 --I 0 I 2 -
I 0 I 1 4 1 0 I 1 3 I 1 0
9
1 - 1 1 1 -
1
Our final goal was to verify Eq.(BSO), which determines the st@le range o tation constant p . Since in our simulation the number of weights of P (z) was n
E [ S 3 = E [ ( u ; ) * ]= 1, the system should be stable for
Refer again to Table B. 1. Comparing measured and theoretical values of weigh various values of p, it is clear that when p = 0.05, the system was theoreticall but experimentally just barely stable. Although the system was still stable, it was with weight noise that suddenly became enormous as p was raised to the stated lev when p was further raised to p = 0.051, the algorithm blew up. Hence, for this case, our prediction of the stable range of p was correct to within 2 percent. This experiment gives one confidence in the formulas and gives one a feeli analysis of scheme C, although based on a few assumptions, is reasonably accurat useful.
SUMMARY
The principal results of this appendix are summarized in Table B.3, which com performance characteristics of modeling schemes A, B, and C.
1
TABLE 8.3
Weight Noise Covariance Matrix
Modeling Scheme B
Modeling Scheme C
PERFORMANCE CHARACTERISTICS OF MODELING SCHEMES A. B,AND C Modeling Scheme A
variance matrices can be compared. The numerator of the covariance for scheme than that for scheme A. With small p, the denominators are about the same. The weight noise for scheme B is larger than for scheme A, while the rate of conve scheme A is as good or better than the rate of convergence of scheme B. Therefore point of view, scheme A gives superior performance over that of scheme B, exc one were not concerned with weight noise, scheme B could be driven a lot harde to converge faster than scheme A. Why else might one use scheme B used instead of scheme A? The answer input to the adaptive model of scheme B is driven by dither, which has contro predictable spectral characteristics and may be much better for modeling than th model input of scheme A, which consists of dither plus the output of the contr output of the controller, the driving input to the plant, is potentially unreliable as a signal; it may be highly nonstationary, it may at times be much stronger than the have poor bandwidth characteristics, and it may not be representable as a stochast all. For example, it might consist of sporadic transient pulses, step functions, an To do a better modeling job and to ensure a wideband fit between the model and scheme B may be the better choice. Another reason for choosing scheme B arises when the plant disturbance is with the plant input. This occurs when plant disturbance obtained at the plant ou tered and fed back into the plant input to achieve plant disturbance canceling. T was discussed in Chapter 8. Scheme C was devised to be an improvement over scheme B. Like scheme C offers a better, more reliable modeling signal than that of scheme A. Both sch scheme B do their modeling based on the dither signal alone. The improvement of over scheme B comes from the elimination of the component y i , shown in Fig. the error ck. This reduces the minimum mean square error and reduces weight n scheme B, scheme C can be used even when the plant disturbance is correlate plant input, as is the case when plant disturbance is canceled. Studying Table B. 1, scheme C may be compared with scheme A. The stab p is the same for both. The weight noise covariance is the same when choosing value of p for both schemes. The time constant for scheme C is larger than all the stants for scheme A, except that with big eigenvalue spread for Ru,, the slowes scheme A will be comparable in decay rate to the single mode of scheme C. The formance in terms of weight noise and speed of convergence is similar, comparing with scheme A, but scheme C has better modeling properties than scheme A. Again referring to Table B. 1, we observe that the stable range of p for sc not as great as that of scheme B, so that, disregarding weight noise, scheme B can
weights with the same speed of convergence. Scheme C is generally the schem It should be noted, finally, that all three schemes, using dither, have the d of disturbing the plant and creating noise at the plant output. This is a “necess many cases when the plant driving function cannot be relied upon as a suitab signal. When studying Figs. 4.3(a), (b), and (c), it is clear that all three schem exactly the same dither noise at the plant output. None of the schemes has an a this regard.
Bibliography for Appendix B
[l] B. WIDROW,“Adaptive filters,” in Aspects of Network and System T R.E. Kalman and N. De Claris (New York: Holt, Rinehart and Winston, 197 587.
[2] L.L. HORWITZ, and K.D. SENNE,“Performance advantage of complex LM
trolling narrow-band adaptive arrays,” IEEE Trans.on Circuits and Systems 28, No. 6 (June 1981), pp. 562-576.
[3] R.R. BITMEAD,“Convergence in distribution of LMS-type adaptive par mates,” IEEE Trans. on Auto. Control, Vol. AC-28, No. 1 (January 1983),
[4] 0. MACCHI,and E. EWEDA,“Second-order convergence analysis of stoc
tive linear filtering,” IEEE Trans. on Auto. Control, Vol. AC-28, No. 1 (Jan pp. 76-85.
A Comparison of t Self-Tuning Regulator blstrom and Wittenmark wi the Techniques of Adapti Inverse Contr
The best-known adaptive control methods are based on the self-tuning regulator of A and Wittenmark. Their 1973 paper [2] has had great influence worldwide in the f adaptive control. Chapter 3 of their book entitled Adaptive Control [ 13 summarize work on the self-tuning regulator. Figure C. 1 is a generic diagram of the self-tuning lator, based on Fig. 3.1 of Adaptive Control. Process parameters
Design
Estimator
Regulator
Process
Figure C.l The self-tuning regulator based on Fig. 3.1 of K.J.ASTROM, and B . W I T T E N M A Conrrol (Reading, MA: Addison-Wesley, 1989).
The system of Fig. C.l is linear and SISO, and it works in the following way process or plant is excited by its input u. Its output is y . This output contains a res
uc
-
Input controller
Figure C.2
m
An alternative representation of the self-tuning regulator.
In Chapter 8, we demonstrated that the adaptive disturbance canceler of F imizes the plant output disturbance power. In fact, we have shown that no othe tem, regardless of its configuration,can reduce the variance of the plant output to a level lower than that of Fig. 8. I . Comparing the self-tuning regulator of F the adaptive disturbance canceler of Fig. 8.1, the question is: Can the feedback c the self-tuning regulator be designed to cancel the plant disturbance as well as t disturbance canceler? Another question that arises is: Can an input controller b for the self-tuning regulator so that when it is cascaded with the plant and its fee troller, the entire control system will have a transfer function equal to the transfer a selected reference model? It is not obvious that the self-tuning regulator and t inverse control system will deliver performances that are equivalent to each oth
C.l
DESIGNING A SELF-TUNING REGULATOR TO BEHAV A N ADAPTIVE INVERSE CONTROL SYSTEM
To address these issues, we redraw Figs. C.2 and 8.1 as Figs. C.3 and C.4, resp order to simplify and bring2ut essential features of these block diagrams. For sim have drawn Fig. 8.1 with P ( z ) approximated by P ( z ) . Also, we included a nec delay z-' within the feedback loop of the self-tuning regulator that will be neces there is no delay either in the plant or in the feedback controller. We will assum is stable. If the plant is not really stable, let it be stabilized by a separate feedbac and let P ( z ) represent the stabilized plant. This creates no theoretical problems tuning regulator, and as demonstrated in Appendix D, creates no theoretical pr adaptive inverse control. To compare the two approaches, we need to first show, that the transfer function from the plant disturbance injection point to the pla
2 Unit delay
Input controller
FC(z)
Feedback controller
Figure C.3
Another representation of the self-tuning regulator.
Plant disturbance N(z)
P o
Input
1
l-$~--j7+-1 Figure C.4
Another representation of the adaptive plant disturbance canceler
For the adaptive disturbance canceler of Fig. C.4, this transfer function is In order for these transfer functions to be equal, it is necessary that
To obtain F C ( z ) , we need Q ( z )and P ( z ) . In practice, both wouldbe readily a a close approximation) from adaptive processes already described. So there w problem in getting a good expression for F C ( z ) .
response of a selected reference model, we multiply the transfer function ((2.4 and set the product equal to M ( z ) :
M ( z ) = ZC(Z) . Z - I
*
P (z) (1 - z-' P ( z ) . Q ( z ) ) . 9
Accordingly,
- M ( z ).FC(z) Z-I
*
P (z)* Q ( z )'
For the entire self-tuning regulator to be stable, it is necessary that ZC(z)be stab has already been established for the rest of the system. M ( z ) is stable. Since P ( are stable, they would not cancel any unstable poles of F C ( z ) that may occur. It i therefore, for F C ( z ) to be stable in order for ZC(z) to be stable, although this is n for stability. I C ( z ) will be unstable if either P ( z ) , Q ( z ) ,or both are nonminim How would one build a self-tuning regulator if its feedback controller F unstable? There are two possibilities. One possibility would be to choose an F stable but not optimal for plant disturbance canceling. The other possibility wou the optimal, unstable F C ( z ) inside the feedback loop and build the input contr having the indicated poles and zeros but allowing components of ZC(z) to b as required for stability, The noncausal filter could be realized approximately propriate delay. The entire system response would be a delayed version of the M ( z ) . These difficulties are not encountered when the optimal F C ( z ) is used a controller ZC(z) is stable.
C.2
SOME EXAMPLES
Specific examples will help to clarify some of the issues. Suppose that the plant is constant, that it is a Dc offset or bias. The question is: How well do the a turbance canceler and the self-tuning regulator handle this disturbance? For t disturbance canceler, the transfer function from the plant disturbance injection plant output is given by (C.2). We would like this transfer function to have a v at zero frequency, that is, at z = 1. This is easily accomplished as follows: 1
1 - P ( 1 ) . Q(1) = 0, or Q(1) = -
W)'
Assuming that the plant transfer function is well behaved at z = 1, it is clear t must be infinite at z = 1. One way to accomplish this would be to let F C ( z ) b integrator, 1 FC(z)= 1---I' ~
giving it a simple pole at z = 1. The input controller has the transfer function give
P ( z ) and Q ( z ) are both stable and finite at z = 1. If in addition M ( z ) has a f at z = 1, I C ( z ) will have a pole at z = 1 and will thereby be unstable making system unstable. A noncausal realization of I C ( z ) would not be useful in this only possibility would be to choose an F C ( z ) having its pole slightly inside the sacrificing some disturbance canceling capability for a stable I C ( z ) . The same kind of result would be obtained if the plant disturbance were a amplitude sine wave. The adaptive disturbance canceler would adapt and lear nate it perfectly. The self-tuning regulator would either be unstable or, if stable, w somewhat less than optimal disturbance reducing performance.
SUMMARY
The self-tuning regulator has an easier job of disturbance reduction and dynam than adaptive inverse control when the plant is unstable with a pole or poles on the or outside the unit circle. Feedback used by the self-tuning regulator has the ca moving the plant poles inside the unit circle to stabilize the plant, reduce distur control its dynamics. For adaptive inverse control, the first step would be to st plant with feedback. The choice of feedback transfer function for stabilization wo critical and would not need to be optimized. The only requirement would be to stabilize the plant. Then adaptive inverse control could be applied in the usual m discussion of initial stabilization for adaptive inverse control systems is given in D. A difficult case for both approaches occurs when the plant has one or mor the unit circle. The inverse controller tries to put the FIR equivalent of a pole or p
a digital integrator. The loop around the cascade would need to be designed to b adaptive inverse control. When the plant is nonminimum-phasewith zeros outside the unit circle, w in Chapters 5,6 and 7 how adaptive inverse control can readily cope with such a ing with this kind of plant with a self-tuning regulator is difficult, much more d dealing with a minimum-phase plant. The literature is not clear on how this ca Comparing adaptive inverse control with the self-tuningregulator reveals one approach is advantageous, and cases where the other approach is advantag there are many cases where both approaches give equivalent performance altho tem configurations and methods of adaptation are totally different.
Bibliography for Appendix C
[ l ] K.J.ASTROM,and B. WITTENMARK, Adaptive control, 2nd ed. (Menlo Addison Wesley, 1995).
[2] K.J. ASTROM,and B.WITTENMARK, “On self-tuningregulators,”Aufoma No. 2 (1973).
‘This situation is not so clear. Refer to simulation examples in Chapter 12.
Adaptive Inverse Contr for Unstable Line SISO Plan
If the plant to be controlled, P ( z ) , is unstable, it is impossible to directly apply ad inverse control to realize a stable well-controlled system. The reason for this is that a forward controller driving an unstable plant will leave the plant unstable. The first step practical utilization of adaptive inverse control for an unstable plant must therefore be lization of the plant. Figure D. 1 shows an unstable plant P ( z >being stabilized by feed The plant disturbance is represented as an additive disturbance at the plant output N feedback filter used within the feedback loop has the transfer function F B ( z ) . The nec delay around the feedback loop is z-' . I Plant disturbance
N W
Feedback filter FB(2) =
-
Unit delay
Figure D.1
Using feedback to stabilize an unstable plant.
'If this delay is not already incorporated in the plant or in the stabilization feedback filter, then a un should be incorporated as shown in the figure. Otherwise it should be omitted.
The input-output transfer function of the feedback system of Fig. D. 1 is given b
-
NP(2). D F B ( Z ) D,(z). D F B ( Z) NFB(zN ) .P ( z )* z-'
'
The objective is to choose F B ( z ) so that H ( z ) will be stable. Once this is
H (z) becomes an equivalent plant whose dynamic response can be controlled
inverse control and whose disturbance can be canceled by the associated adap bance canceling techniques. The roots of the denominator of (D.3) need to be placed inside the unit c . question is, how doe z-plane by appropriate choice of N,cB(z) and D F B ( z ) The this appropriate choice? Is the choice unique? Should the choice be optimal? O wide latitude in this process? To begin, let the choice be arbitrary as long as H ( Furthermore choose the feedback filter F B ( z ) to be stable.
D.l
D Y N A M I C CONTROL OF STABILIZED PLANT
Consider first the issue of dynamic control of H (z). This is the same issue as dy trol of the plant P ( z ) , since the plant output is really the same as the output of need to examine the model-reference inverse of the transfer function H ( z ) . Usin
For sake of argument, imagine the inverse controller as being a cascade o filters, each having a transfer function that closely approximates the first, secon factors of the right-hand side of equation (D.4) respectively. Since F B ( z ) has b to be stable, the second factor will be stable since its denominator D F B ( Zhas ) all side the unit circle. A filter whose transfer function closely approximates the se is therefore easy to realize as a causal filter without modeling delay. A filter wh function closely approximates the first factor is easy to realize if N p ( z ) has all side the unit circle in the z-plane. If not, realization requires modeling delay. In ease of realization of the first factor depends on plant characteristics and will b regardless of the choice of F B ( z ) . Realization of the second factor will always b F B ( z ) was chosen to be a stable filter and one that stabilizes H ( z ) , but otherw pletely flexible. Realization of the third factor will be easy because M ( z ) wil chosen to be stable.
I Command input Controller Stablizing filter
I
I I
Unit delay
Modeling
I I I I
I I I I I I
Figure D.2
I
-
A -
I-4Tl-J
Reference model
I I I I
I I
I I I I I I
Inverse control of dynamics of unstable plant.
To design the filter F B ( z ) , some idea of the plant characteristics would Then root-locus or Nyquist design of this plant stabilizer could be done. The de critical, and it could be chosen by experiment. The above arguments show that i lizing filter is itself stable,z wide latitude in its design is acceptable. Variation in will result in variation in C(z& but the overall system dynamic response will alw same. Ease of realization of C(z) will not depend on the choice of FB(z). The next issue is plant disturbance canceling. Since stabilization of the pl essary, a question arises about the optimality of the stabilization approach of Fig
P ( z ) stabilized by feedback. A feedback filter with transfer function F B ( z ) prov bilization signal. An adaptive process using the plant driving signal V Q )is u tively find H ( z ) , an estimate of H ( z ) given by Eq. (D.3). A copy of H ( z ) is tain the disturbance at the plant output. This disturbance is filtered by Q ( z )an subtracted from the plant input to cancel the plant disturbance as well as it can squares sense. We would like to show that adaptive disturbance canceling with an unstab be optimal regardless of the choice of the feedback filter F B ( z ) , as long as it s plant. We will prove that this is true if the plant is minimum-phase and the fee is chosen to be stable. Please refer to Chapter 8, particularly Section 8.2, “Proof of Optimality fo tive Plant Disturbance Canceler,” and Fig. 8.2. Many of the mathematical ideas tion will be useful here. Figure D.4 shows a collection of block diagrams that , optimal transfer function for the disturb obtaining an expression for Q * ( z ) the ing feedback. The filter Q ( z ) in Fig. D.3 adapts by an offline process toward solution. Of interest is the development of expressions for Q * ( z ) and for the p uncanceled plant disturbance. Figure D.4(a) represents the unstable plant, P ( z ) , and its additive statio disturbance N ( z ) . Figure D.4(b) again shows P ( z ) with its output disturbanc here the disturbance is represented as if it originated from a unit-power white L ( z ) which has gone through a stable, causal, minimum phase filter r ( z ) . The analogous to that of Fig. 8.2(b). Figure D.4(c) shows the plant P ( z ) now bein with feedback. The output disturbance is no longer N ( z ) . It is equivalent to
In Fig. D.4(c), the plant output disturbance is generated by L ( z ) propagating filter Y ( z ) .The transfer function of this filter is
Figure D.4(d) is a simplified representation of Fig. D.4(c). Figure D.4(e) show of the actual plant disturbance canceler. The equivalent plant disturbance drive bance canceling path consisting of the delay z-’ , the filter -Q*(z), and the trans of the stabilized plant H ( z ) . The optimal transfer function - Q * ( z )minimizes t the plant output disturbance after cancelation. The block diagram of Fig. D.4(e like that of Fig. 8.2(d). We need an expression for Q * ( z ) .The Shannon-Bode method will be us it. The causal Wiener filter is given by Eq. (2.74).
I
I
Figure D.3
I
An adaptive noise-canceler for an unstable plant.
Whk%ise, unit power
L(z) White noise, unit power
A-1
Stable, causal minimum-phase
rw
(d) Figure D.4
Calculation of plant output disturbance after cancelation. (continued on next
U Z )
White noise, unit power
Wz) Z-’
(0 Figure D.4
(continued).
Figure D.4(f) is equivalent to Fig. D.4(e), except that the order of filtering b Q * ( z )has been commuted. Since Q * ( z )is the Wiener filter, we make use of o Wiener filter notation and designate the input to Q * ( z )as F ( z ) (fk in the time do we designate its desired response as D ( z ) (4 in the time domain). From Q. Wiener solution is
Now, referring tc Fig. D.4(f), we note that
We will need to factor @ f f ( z )as @ff(Z)
= q z ) * @&z).
To accomplish this in a useful way, we will factor r ’ ( z ) in the following manne
r’(z)= r;w . r;w.
Let the factor r; (2) have all the zeros and poles of P ( z ) that are inside the unit the factor r ; ( z ) have all the zeros of r’(z)that are outside the unit circle. It h because r ’ ( z )has no poles outside the unit circle, a fact that can be verified by of Eqs. (D.3) and (D.6) as follows. The factor r ( z ) of (D.6) is stable and minim
the choice of the feedback filter F B ( z ) . Referring to Q. (D.8), factorization of @ J J ( Z ) also requires factorizatio H ( z - ' ) . To do this factorization, we first note that N ( z ) is stable and, from ( be deduced that H ( z ) is minimum-phase. The zeros of H (z) correspond to the plant and to the poles of F B ( z ) . We now assume that the plant P ( z ) has all inside the unit circle. Since F B ( z ) is stable with all of its poles inside the unit is minimum-phase and
H ( z ) = H + ( z ) and H ( z - ' ) = H - ( z ) .
Factorization of Q f f ( z ) proceeds in the following way. Substituting (D.11) int
O n ( z ) = r; ( z ) r ; ( z ) r ;( z - ' ) r ; ( z - l ) H ( z ) H( z - ~ ) .
Grouping the poles and zeros inside the unit circle and, respectively, those out circle yields
and then substitute this and (D.14) and (D.15) into (D.7). The result is
To evaluate the bracketed factor of (D.17), it is useful to note that r ; ( are both stable and causal and that their product is stable and causal. This pro expanded therefore as follows.
The bracketed factor of (D.17) may now be evaluated as
transfer function. We need to find this disturbance transfer function. From Fig. inspection, disturbance transfer function
)
=
r’(z)(1 - Q*(Z). z - I
HW).
Substitution of (D.20) into (D.21) gives
Substitution of (D.11) into the above yields
The uncanceled plant disturbance is therefore equivalent to white noise of propagating through this transfer function. We are on the verge of a very useful result. We will show next that the d transfer function (D.23) does not change as the stabilization feedback filte changed. This transfer function is a function of the plant characteristic, but it tu be a function of F B ( z ) . Regarding (D.23), the factor r;(z),defined by (D. 11) and (D.6), has been established as dependent on P ( z ) but independent of the choice of F B ( z ) . The s of course for the denominator factor of (D.23), r;(z-l).All that remains is a que yo: Does it depend on F B ( z ) ? Equation (D.18) defines yo as the first term in the expansion of T’(z) in a ries of z - l . The first term can be found by evaluating expression (D.6) as z-I result is
or
lizes the plant P ( z ) . We have also assumed that P ( z ) is minimum-phase. Furt has been performed with the minimum-phaserestriction on P ( z ) relaxed. The ze were allowed to be either inside or outside the unit circle or some inside, some o analysis is straightforward, but too lengthy to include here. The conclusion is th output disturbance power; after adaptive cancelation, is independent of FB(z FB(z) is itself stable and is able to stabilize the plant P(z). There are no res P ( z ) except that it should not have zeros on the unit circle.2 We have not proved that linear feedback stabilization of the plant, exemp diagram in Fig. D.l, is optimal. We simply do not know of any other way to s unstable plant. Given that the plant is stabilized as in Fig. D.l, minimization o turbance proceeds from there. We have shown that the choice of the feedbac long as it stabilizes the plant and it alone is stable, is not critical to the disturba ing process. All such feedback filters would perform equivalently from the po of disturbance canceling. The minimum plant output disturbance power would upon the choice of FB(z). Recall that in thejrst section of this appendix, we s the ability of an adaptive inverse controller to control plant dynamics does not a on the choice of FB(z),as long as it alone is stable and it is able in a feedback to stabilize the plant. Adaptive inverse control can be made to work with an unstable plant as w a stable one, as long as the unstable plant isfirst stabilized withfeedback. The d feedback is not critical. Control of system dynamics is optimized separately from plant disturbance. Optimization of system dynamics does not interfere with min plant disturbance.
0.3
A SIMULATION S T U D Y OF P L A N T DISTURBANCE CANCELING: A N UNSTABLE PLANT WITH STAB ILI2AT I0N FEEDBACK
In order to demonstrate that the choice of FB(z),the stabilization feedback, doe the ability of an adaptive canceler to cancel plant disturbance (as long as FB stable), the system of Fig. D.3 was simulated. An unstable plant P ( z ) was chose different feedback filters, FBi(z), i = 1, 2 , 3, each capable of stabilizing the chosen.
ZEventhis restriction can be circumvented. If, for example, the unlikely case occurred that the p at z = 1, the plant could be preceded by an integrator having the transfer function (1 - z - ’ ) - ’ .
For each i = 1,2,3, we obtained a different transfer function H i ( z ) and disturbance canceling loop transfer function Qi ( z ) . However, as expected, the po uncanceled plant disturbance remained the same in all three cases. In all of these simulations, the unstable plant was 1 P(z) =
m,
The plant disturbance N ( z ) was simulated by filtering a zero-mean white noise low-pass 100-weight filter having the transfer function 1 +z-I
+ .. . + z - .
The power output was normalized so that
E [ n i ] = 1.
The plant model E ( z ) was obtained from H ( z ) but truncated to have 10 and the disturbance canceling loop had a Q ( z )limited to 50 weights. The three plant stabilization filter were F B , ( z ) = -1.5, FBz(z) = -2, and FB3(z) = - (1
1 + $).
The result of the simulation experiments are given in Table D. 1. In colum table, the transfer functions of the stabilizationfilters FBi ( z ) are given. In column functions of the equivalent plants H i ( z ) (i.e., overall transfer functions of the with corresponding stabilization feedback, as obtained from Eq. (D.3)), are give one, the equivalent plant had one pole, for case two, the equivalent plant had no case three, the equivalent plant had two poles. In column 4 of Table D. 1, the transform of the output disturbance of the plant is given for all three cases. This equivalent disturbance appears at the ou plant stabilization feedback system shown in Fig. D. 1. These transforms take in the impact of the feedback filter F B ( z ) , in accord with Eq.(D.5). Using the offline adaptive process depicted in Fig. D.3, we have computed t 50-weight disturbance canceling filter Q ( z ) .In each cas2 we have assumed that given in column 3, except that it is truncated to allow H ( z ) to have only 100 w finding Q ( z ) ,the modeling signal was designed to have the same spectral chara equivalent plant disturbance given in column 4 of the table. Although Q ( z ) had 5 only the first 8 weights for each case are listed in column 5 of the table. Finally, using Qi ( z ) as obtained above, we ran the system of Fig. D.3 for For each case, we measured the power of the uncanceled plant disturbance. Resu
Equivalent plant noise output 4 5
Equivalent plant H ( z ) 3
Noise canceling feedback Q(z) truncated to its first 8 weights
0
0.02825
output disturbance power 6
ESULTS OF DISTURBANCE CANCELING EXPERIMENTS WITH AN UNSTABLE DISTURBED PLANT STABILIZED BY DBACK FILTERS
back
FB(z) 2
0
0
0.02819
0.02841
0.0039
0.0154 0.0077 0.0039
0
0.9933 -0.496
0.0308
0
N ( z1)( l f-2z-') z-I
0
I
fz-'
0.1234 0.0617 0.0308 0.0154 0.0077
1
1-
0.4935 0.2467
1.6168 -0.435
2
.5
N(z)(l-2z-')
1.4935 -2.2402
I-z-l+fZ-2
N ( z ) ( l -2z-1
I I-z-'+fZ-2
1
2
to it (i.e., G(z)has enough weights to form a good model of H ( z ) ,and the adap stant is small enough so that weight disturbance in the plant model is negligibl both depend on the disturbance cancelation feedback filter Q ( z )to have enough achieve good cancelation of the plant disturbance. But what happens if these a are incorrect? Suppose that the length of the model g ( z ) were limited. If the stabilizatio filter were chosen so that the impulse reskonse of the stabilized plant H ( z ) wo atively short, then limiting the length of H ( z ) would not cause errors. Howev that the stabiljzation feedback filter were chosen so that the impulse response of long. Then H ( z ) would be too short to adequately model H ( z ) . The result would power of the uncanceled plant disturbance would grow. In this sense, the ability turbance canceler to cancel plant disturbance could depend on the choice of the st feedback filter. In order to demonstrate this possibility, we reduced the number of weig to only five. Then we repeated the entire simulation for all three cases. In case H ( 2 ) is very simple), the uncanceled plant disturbance was essentially unaltered had the value 0.03020. In case one, H ( z ) is more complicated with a longer sponse, and limiting the length of H ( z ) had an effect on the uncanceled plant power, which grew to 0.0653. In case three, H ( z ) is even more complicated w impulse response, so that limiting E ( z ) had an even greater effect, causing the plant disturbance power to shoot up to 0.403. Similarly,one can expect that limiting the number of weights in the distu celing feedback filter Q ( z )will introduce some sensitivity to the choice of the s feedback lilt?. To explore this issue, the simulation was rerun with different len filter Q ( z ) . H ( z ) was restored to its original length of 100 weights. With Q ( z ) set to have 10 weights, the three cases were run again. The sp resulting uncanceled plant disturbance powers varied over a range of 1 percent 0.7 percent for the 50-weight Q ( z ) ) . This is not a significant variation. Fina periments were rerun with Q ( z ) limited to only 3 weights. For cases one and t canceled plant disturbance power remained essentially the same, but for case t more complicated H ( z ) ,the uncanceled plant disturbance power jumped up by This is significant. Therefore, with limitations placed on the length of the impul of Q ( z ) ,we note once again that the ability of the disturbance canceler to cance turbance could depend on the choice of the stabilization feedback filter.
D.5
SUMMARY
It has been demonstrated above that use of the system of Fig. D.2 will allow ada control of overall dynamic response when involving an unstable plant P(z) w tion feedback, just as is done with a stable plant. P ( z ) may have some zeros in circle in the z-plane and may have some zeros outside it. The stabilized equ H(z) will have the same zeros. Additional zeros of H ( z ) come from the poles lization filter FB(z). If kB(t)is stable, then H ( z ) will have no more zeros out circle than P(z) had at the outset. The difficulty of making an inverse filter relates primarily to the number side the unit circle of the filter to be inverted. From the above arguments, we m that the choice of FB(z) is not critical to the process of using inverse control to c all system dynamics, as long as FB(z) is stable by itself, as long as it stabilizes th within the stabilization loop, and doesn’t give H ( z ) any more zeros outside th than the ones originally had by P(z). The system of Fig. D.3 can be utilized for canceling the plant disturbanc plants. Performance of the scheme will be optimal and insensitive to the choic bilization feedback filter FB(z) as long as this filter is stable by itself. Howeve advantageous to design the stabilization feedback, if possible, so that the impu of the stabilized plant would not bezxcessively long, thus making H ( z )easy to erwise, the impulse responses of H(z) and/or Q(z) would need to be long to performance. We conclude that systems of Figs. D.2 and D.3 are indeed capable of ada control and adaptive disturbance canceling with unstable plants. The procedure following: Find a stable filter F B ( z ) that is capable of stabilizing P(z) within t tion loop of Fig. D. 1. Then treat the stabilized plant H ( z ) like any other stab apply the standard methods of adaptive inverse control for dynamic response a turbance.
Orthogonalizing Adaptiv Algorithms: RLS, DFT/LM and DCT/LM
We saw in Section 3.6 that the mean square error of an adaptive filter trained with the algorithm decreases over time as a sum of exponentials. The time constants of the ponentials are inversely proportional to the eigenvalues of the autocorrelation matrix inputs to the filter. This means that small eigenvalues create slow convergence modes MSE function. Large eigenvalues, on the other hand, put a limit on the maximum le rate that can be chosen without encountering stability problems (see Section 3.4). It from these two counteracting factors that the best convergence properties are obtained all'the eigenvalues are equal, that is, when the input autocorrelation matrix is proportio the identity matrix. In this case, the inputs are perfectly uncorrelated and have equal p in other words, they are samples of a white noise process. As the eigenvalue spread of the input autocorrelation matrix increases, the c gence speed of LMS deteriorates. As a matter of fact, it is not unusual to find practic uations where the inputs are so correlated that LMS becomes too slow to be an acce way to go (see Chapter 7 for an example). To solve this problem, a variety of new rithms have been proposed in the literature. Their actual implementations and prop are all different but the underlying principle remains the same: trying to orthogona much as possible the input autocorrelation matrix and to follow a steepest-descent p the transformed error function. These algorithms can be divided in two families: First, the algorithms such a (recursive least squares) and recursive least squares lattice filters that extract inform from past data samples to decorrelate present input signals, and second, the algorithm as DFTLMS and DCTLMS (discrete Fourier and discrete cosine transform - least squares), which attempt to decorrelate the inputs by preprocessing them with a transf tion (the DFT or the DCT) that is independent of the input signal.
'By Franloise Beaufays, Department of Electrical Engineering, Stanford University, and SRI Intern Menlo Park,CA.
the weight vector using the so-called matrix inversion lemma (see below). The two main differences with LMS are that an estimate of the autocorrel is used to decorrelate the current input data, and that, at least for stationary inputs of the steady-state solution keeps on improving over time, eventually leading solution with no misadjustment. The major disadvantages of RLS over LMS are its poor tracking ability tionary environments [4], its complexity (due to matrix by vector multiplicat operations are required for each weight update whereas only O(n)operations ar with LMS), and its lack of robustness (the algorithm can be unstable under c conditions). The computational cost and robustness issues have been addressed by re developing other exact least squares algorithms. The lattice filter algorithms, f require only O ( n )operations per iteration and are far more robust (although it h served that finite precision effects can severely degrade the algorithm perform The reduction in number of operationscomes from the fact that Toeplitz autocom trices have only n distinct elements. By making a one-to-onecorrespondence be n elements and n so-called rejection coeflcients, and by manipulating the algeb putational cost can be reduced from O(n2)to O(n). The price for this improv increased complexity of the algorithm in terms of number of equations to be im number of variables to be stored, and general complication of the algebra. In this appendix, we will limit ourselves to presenting RLS, the simple basic of the exact least squares algorithms. For a detailed description of other we refer the reader to Haykin's textbook on adaptive filters [ 101.
E.1.2
Derivation of the RLS Algorithm
To avoid instability problems, the error function to be minimized in recursive l is generally defined as an exponentially weighted sum of squared errors [lo]: k
i=l
where fl 5 1 is a positive constant generally chosen close to one, and ei is the at time i . At time k, the best estimates of R and P are given by
where Kk, the gain vector, is defined by
and
Introducing (E.3) and (E.6) in (E.4), and after some algebraic manipulation, weight update formula
where
Note the formal resemblance with LMS. The weight vector is updated prop to the product of the current input xk and some error signal f f k . The error in RLS a priori in the sense that it is based on the old weight vector wk-I, whereas in error ek = dk - W:Xk is computed a posteriori, that is, based on the current wei A more important difference though is that the constant learning rate p of LMS by a matrix Gk that depends on the data and on the iteration. In that respect, R thought of as a kind of LMS algorithm having a matrix-controlled optimal lea The weight update formula could also be rewritten as
Wk = Wk-l
+ kk RL!lffkXk,
where kk is a time and data dependent scalar learning rate equal to kk =
p-'
1 + B-' X r RL!l
Xk
'
*Let A and B be two positive definite N x N matrices: C an N x M matrix, and.Da positive def matrix. The lemma says that if A = B+CD-'C', then A-' = B-' - B-'C(D+C'B-'C)-'C'B-'.
It also suffers from possible numerical instability and from bad tracking perform stationary environments when compared to LMS. LMS is intrinsically slow bec not decorrelate its inputs prior to adaptive filtering but preprocesses the inputs mate of the inverse input autocorrelation matrix in the fashion of RLS, and this problems cited above. The solution we propose in the next section consists of pr the inputs to the LMS filter with a fixed transformation that does not depend o input data. The decorrelation will only be approximative, but the computation remain of O(n) and the robustness and tracking ability of LMS will not be affe
.2
T H E D F T / L M S A N D D C T / L M S ALGORITHMS
E.2.1
Introduction
The DFTLMS and DCTLMS algorithms (discrete Fourier/cosinetransform squares) are composed of three simple stages (see Fig. E. 1.). First, the tap-del are preprocessed by a discrete Fourier or cosine transform. The transformed sign normalized by the square root of their power. The resulting equal power signa to an adaptive linear combiner whose weights are adjusted using the LMS algo these two algorithms, the orthogonalizing step is data independent; only the pow ization step is data dependent (i.e. the power levels used to normalize the sign mated from the actual data). Because of the simplicity of their components, these retain the robustness and computational low cost of pure LMS while improving gence speed. The algorithm described in Fig. E. 1 is summarized in Table E. 1 . 3 The speed improvement over LMS clearly depends on the orthogonalizi ties of the transform used. No general proof exists that demonstrates the superi transform over the others; many other transforms than the DFT and the DCT co sidered [ 111 (for example, the discrete Hartley transform [S] or the Walsh-Hada form [ 15,7]). The DFTLMS, first introduced by Narayan [ 121, is perceived as t algorithm of the family because of the exponential nature of the DFT and bec strong intuition scientists have developed about the Fourier transform. It is our though that in most practical situations DCTLMS performs much better than
31n practical applications, the power normalization on the inputs is often replaced by a power nor the learning rate of the LMS algorithm. The resulting algorithm is more robust when the input powe noisy[l]. In the remainder of this appendix, we will assume that the power estimates are noise fre the two algorithms are identical.
X
k
Error
ek
Figure E.l
1-@
dk
Desired Response
DFT/LMS and DCTLMS block diagram.
with K , = l / f i for i = 0 and I otherwise. Power Normalization uk(i) = u l i ( i ) / J % i G where E is a small constant. Pk(i)=y P k - i ( i ) + ( ~ - - y ) u ; ( i ) y E 10, 11 is generally chosen to be close to 1.
V i = O , ...,n V i = O , ...,n -
LMS Filtering . .
Vi=O, ....n -
w t + i ( i ) = w ~ ( i ) + p e e t u;(i) where u; is the complex conjugate of uk.
and other similar algorithms. In addition, it has the advantage over DFTLM valued. Nevertheless, we will center our explanations on DFTLMS for reasons ity, although for practical purposes (cf. for example Sections 7.9 and 7.10) we use the DCTLMS. In the remainder of this section, we will, through different approaches, e itively and analytically the mechanism of the proposed algorithms. We will t some theoretical results comparing the performances of LMS, DFTLMS, and and we will conclude with a short discussion of computational cost.
E.2.2
A Filtering Approach
The DFT-DCT performs essentially n linear transformations from the input v (xk,xk-I, ..., ~ k - , + l ) ~ tothenoutputs,Uk(O),U~(l),...,uk(n- 1). Eachtransf characterized by an impulse response h; ([) = T,(i, I ) . The corresponding trans is given by "-I
For the DFT, Hi(w) is equal to
/=O
which represents a bandpass filter of central frequency 2rriln. The DFT can th as a bank of bandpass filters whose central frequencies span the interval [0,2rr] shows the magnitude of a sample transfer function for a 32-point DFT. At each
1 o.6
t
1O-id
Figure E.2
1
2 3 4 5 Normalized frequency w E [O, 2x1
6 2
Magnitude of a sample transfer function for a 32 x 32 DFT, IHs(w)12
For the DCT, the ith transfer function is equal to
The Hi(o)'s still represent a bank of bandpass filters but with different central f different main lobes and side lobes, and different leakage properties. Figure E.3 magnitude of a sample transfer function for a 32-point DCT. The DCT filter loo ferent from the DFT one. In particular, the presence of two symmetrical peaks a single main lobe as in the DFT) comes from the cancelation of the two zeros nominator of Hi ( w ) with two zeros of its numerator. More fundamentally, it is a sequence of the cosinusoidal nature of the transform. Although the DCT does n frequencies the way the DFT does, it is a powerful signal decorrelator as will strated in the next sections.
Y
$ -
3.-
c)
C
$ lo4 lo-‘
oB
1
10.’
Figure E.3
E.2.3
1
2 3 4 5 Normalized frequency w E [0,2n]
6
!
Magnitude of a sample transfer functions for a 32 x 32 JXT, IHlo(w)l
A Geometrical Approach
The DFTLMS and DCTLMS algorithms can also be illustrated geometrical proach does not lead to any practical theory but helps understanding conceptu erations performed by the algorithms block-diagrammed in Fig. E. 1. The DF matrices defined in Table E. 1 are unitary matrices (i.e., their rows are orthogona other and have Euclidean norm one). Unitary transformations perform only ro symmetries; they do not modify the shape of the object they transform. Looking at the MSE formula
((W)
={,,,in
+ (W - W*)TR(W- W*)
given in Section 3.3, and fixing ( to some constant value, we see that (E.17) r hyperellipsoid in the n-dimensional weight space. A unitary transformation o rotates the hyperellipsoidand brings it into approximatealignment with the coor The slight imperfection in alignment is primarily due to leakage in the transfor DFT. The idea is illustrated for a simple two-weight case in Fig. E.4. Figure E the original MSE ellipse, Fig. E.4(b) shows it after transformation by a 2 x 2 D The shape of the ellipse is unchanged and so are the eigenvalues of R. The power normalization stage (cf., Fig. E. 1) can be viewed geometricall formation that, while preserving the elliptical nature of ((W), forces it to cross ordinate axes at the same distance from the center. This operation is not unitary modify the eigenvalue spread. It almost always improves it. The better the al
Figure E.4
E.2.4
MSE hyperellipsoid (a) before transformation, (b) after DCT, (c) after power norm
An Analytical Approach
In order to find precise information on how well a given transform decorrela classes of input signals, one must set the problem in a more mathematicalframew forming a signal Xk = (xk, xk-1,. . . , X k - n f ~ ) ~by a matrix T, (the DFT or the trix), transforms its Toeplitz autocorrelation matrix R = E[XkXF] into a non-To trix B = E[T, XkXF T t ] = T n R T t (the superscript H denotes the transpose c Power normalizing T,Xk transforms its elements (T,Xk)(i) into ( JPowerof(T,Xk)(i) , where the power of (T,Xk)(i) can be found on the mai of B. The autocorrelation matrix after transformation and power normalization
S = (diag B)-'/2 B (diag B)-'/2.
If T decorrelated Xk exactly, B would be diagonal and S would be the iden I and would have all its eigenvalues equal to one, but since the DFT and the D
perfect decorrelators, this does not work out exactly. Some theory has been de the past about the decorrelating ability of the DFT and the DCT (see, for exam 141) but the results presented in the literature are in general too weak to allow anything about the magnitude of the individual eigenvalues of S, which is our ma For example, it has been proven that the autocorrelation matrix B obtained after the DCT asymptotically converges to a diagonal matrix, asymptotically meanin
First-order Markov signals are a very general, practical, and yet simple class They result from white noise passing through a single-pole lowpass filter. Such an impulse response that decreases geometrically with a rate p given by the fi Markov-1 input signal xk = ( x k , . . . , x ~ - , ~ + ~of) ' parameter p E [O, 11 h correlation matrix equal to
For n large (theoretically for n tending to infinity), the minimum and maxi values of an autocorrelation matrix R are given by the minimum and maximum o spectrum of the signal that generated this autocorrelation [9, 81. This result is a sequence of the fact that R is Toeplitz. It can easily be checked that in our cas spectrum of xk is given by p ( w )=
C p'e-j"' +oo
I=-00
=
1 1 - 2p cos(w)
+ p2.
Its maximum and minimum are respectively 1 /( 1 - p ) 2 and I / ( 1 spread of R thus tends to n+m
+ P ) ~ The .
Eigenvalue spread before transformation
This eigenvalue spread can be extremely large for highly correlated signals ( p The autocorrelation S of the signals obtained after transformation by the DCT and after power normalization is not Toeplitz anymore, and the previous th be applied. The analysis is further complicated by the fact that only asymptoti eigenvalues stabilize to fixed magnitudes independent of n, and that power norm a nonlinear operation. Successive matrix manipulations and passages to the lim us to prove the following results (see [ I , 31 for a complete derivation): lim Eigenvalue spread after DFT It'O
(
4Tw0 matrices converge to one another in a weak norm sense when their weak norms converge t The weak norm of a matrix A is defined as the square root of the arithmetic average of the eigenva
of the DCT as a signal decorrelator.
E.2.6
Computational Cost of D F T / L M S and DCT/LMS
In addition to their fast convergence and robustness, DFTLMS and DCTLM advantage of a very low computational cost. The inputs X k , X k - 1 , . . . ,Xk-n+l bei samples of the same signal, the DFT/DCT can be computed in O ( n ) operation DFT, we have
The U k ( i ) ’ S can thus be found by an O(n)recursion from the Uk-1 (i)’s. This type sometimes called s l i d i n g - D F T . A similar O ( n )recursion can be derived with mo for the DCT xi
u f c T ( i ) = uf-yT(i)cos(-) n u f S T ( i )= uk-,
n
n
- (-l)ixk-n
n
- (-l)ixk-n
u f c T ( i )is the ith output of the DCT, u f S T ( i )is the ith output of a DST (di transform) defined exactly like the DCT but replacing “cos” by ‘‘sin’’ (interlaci cursions is necessary and comes basically from the fact that cos(a + b) = cos(a sin(a) sin(b)). Other O ( n ) implementations of the sliding-DFT and the sliding be found in the literature. In particular, the so-called LMS spectrum anafyzer [ 16 shown to implement the sliding-DFT in a very robust way, even when implemen precision floating point chips [2]. The power estimates P k ( i ) are also computed with simple O(n) recursio ble E.l). Finally, the last step: The LMS adaptation of the variable weights, is overall algorithm is thus O(n). For the most part, the DCTLMS algorithm is superior to the DFTLMS Both are robust algorithms, containing three robust steps: transformation, pow ization (like automatic gain control in a radio or TV), and LMS adaptive filter algorithms are easy to program and understand. They use a minimum of comput slightly more than LMS alone. They work almost as well as RLS but don’t have
formation Systems Lab., Stanford University, Stanford, CA, 1994, June 1
and B. WIDROW,“On the advantages of the LMS spectru [2] F. BEAUFAYS, over non-adaptive implementations of the sliding-DFT,” IEEE Trans. on C Systems, Vol. 42, No. 4 (April 1995), pp. 218-220.
[3] F. BEAUFAYS, and B. WIDROW,“Transform domain adaptive filters: cal approach,” IEEE Trans. on Signal Processing, Vol. 43, No. 2 (Febr pp. 4 2 2 4 3 1.
[4] N. BERSHAD, and 0. MACCHI, “Comparison of RLS and LMS algorithm ing a chirped signal,” in Proc. ICASSP, Glasgow, Scotland, 1989, pp. 896
[5] R. N. BRACEWELL, The Hartfey transform (New York: Oxford Unive 1986).
[6] G. F. FRANKLIN, J. D. POWELL, and M. L. WORKMAN, Digital control systems, 2nd ed. (Reading, MA: Addison-Wesley, 1990).
[7] R. C. GONZALEZ, and P. WINTZ,Digital imagepmcessing, 2nd ed. (Re Addison-Wesley, 1987).
[8] R. M. GRAY,“Toeplitz and circulant matrices: 11,” Technical Report 65 mation Systems Lab., Stanford University, Stanford, CA, April 1977.
[9] U. GRENANDER, and G. SZEGO,Toeplitzforms and theirapplications, 2n York: Chelsea Publishing Company, 1984).
[ 101 S. HAYKIN, Adaptive filter theory, 2nd ed. (Englewood Cliffs, NJ: Pre
1991).
[ 113 D. F. MARSHALL, W. K. JENKINS, and J. J. MURPHY, “The use of ortho
forms for improving performance of adaptive filters,” IEEE Trans. on C Systems, Vol. 36, No. 4 (April 1989), pp. 474-484.
[12] S. S. NARAYAN, A. M. PETERSON, and M. J. NARASIMHA, “Transfo LMS algorithm,” IEEE Trans. on Acoustics, Speech, and Signal Processin No. 3 (June 1983), pp. 609-615.
[ 131 R. C. NORTH,J. R. ZEIDLER, W. H. Ku, and T. R. ALBERT,“A floating
metic error analysis of direct and indirect coefficient updating technique tive lattice filters,” IEEE Trans. on Signal Processing, Vol. 41, No. 5 (M pp. 1809-1823.
A MIMO Application: A Adaptive Noise-CanceI System Used for Bea Control at the Stanford Line Accelerator Cent
F.l
INTRODUCTION
A large number of beam-based feedback loops have recently been implemented at of stations along the Stanford Linear Accelerator Center (SLAC) in order to main trajectory of the electron beam down its 3-km length to an accuracy of 20 microns. Si accelerator parameters vary gradually with time, we have found it necessary to use a methods to measure these parameters to allow the feedback system to perform opt Eight beam parameters are being controlled at each station. We therefore have a s eight-input eight-output MIMO systems. The next section gives a general description of the accelerator. Sections F.3 explain the functioning of the steering feedback loops and the need to have the loop municate with each other. Sections F.5 and F.6 describe the theoretical background a ulations used in designing the adaptive system and Section F.7 relates our experienc the system implemented on the actual accelerator.
F.2
A GENERAL DESCRIPTION OF THE ACCELERATOR
SLAC is a complex of particle accelerators located near (and operated by) Stanfo versity. It is a government facility whose primary goal is the pursuit of elementary p physics: the study of the smallest particles of matter and the forces between them. 'By Thomas M. Himel. Stanford Linear Accelerator Center, Stanford, CA.
396
Arcs
e-
3 km Linear Accelerator
Figure El dots.
4
Diagram of the Stanford Linear Collider with locations of steering feedback loops depict
The SLC is a pulsed accelerator. On each pulse a bunch of 3 x 10" posi electrons) is accelerated down the linear accelerator (Linac) from the bottom to th figure. All those positrons are contained in a cylindrical volume 150 microns in 1 mm long. The positrons are followed by a similar bunch of electrons 20 meter takes these particles 10 microseconds to reach the end of the Linac. There the tw follow separate paths. Magnetically deflected, the electrons go to the left and th go to the right. They head around the arcs, a set of a thousand bending and foc nets, which are used to direct the beams toward a head-on collision. Just befo they are focused by superconducting quadrupole magnets down to a radius of a crons (comparable to the transistor size on integratedcircuits). The beams pass th other. For a small fraction of the pulses, an individual electron and an individu will collide head on and convert their energy into matter. This matter is likely form of the neutral intermediate vector boson, the Zo. There is a special detec the beam crossing point which measures the decay fragments of the Zo to try to about this elusive particle and the weak force for which it is the intermediary. T the Z" is one of the major goals of the SLC.
The heart of the SLC is the original 3-km-long linear accelerator which has ily refurbished over the years by giving it a new control system, new focusing m doubling its energy. The Linac is used to simultaneously accelerate a bunch and a similar bunch of positrons so that each particle has an energy of about 47 or billion electron volts). A few examples will aid in the understanding of thi energy. 0
0
0
If you were to put an ordinary 1.ti-volt D-cell battery between two parallel and set an electron near the negative plate, it would be accelerated to the po and end up with an energy of 1.5 eV. To get to 50 GeV, you would need a s teries five times the distance to the moon.
Einstein’s theory of special relativity says matter and energy are different f same thing and relates them with the famous equation E = mc2. At rest, has a mass about 1/2,000 that of a proton. At the end of the SLC, its en sponds to a mass of about 50 times that of a proton.
Special relativity also dictates that nothing can go faster than the speed particles are given more and more energy their speed just gets closer and c limit. At the end of the SLC an electron has a velocity of 99.999999995% of light. This is so close that if the electron beam were to race a beam of moon, the light would only be 2 cm ahead on arrival.
The key to obtaining such a high energy in such a short distance is that the an oscillating electromagnetic field instead of a DC electric field. The field is c a cylindrical copper waveguide that has a 9-mm radius hole down its center. T magnetic wave has a wavelength of 10.5cm and the particles “surf” on it. The is carefully designed so the wave travels at the speed of light; hence, the electro stantly accelerated near the crest of the wave. Creating the electromagnetic wave are 240 klystrons evenly spaced alon Each klystron is a vacuum tube about 6 feet high and is similar in concept to to make radar waves and to power radio transmitters. At SLAC these tubes ar power. Each one produces 50 megawatts although only for a 3.5-microsecond p to their high power output, these tubes are fairly temperamental, break down f and need frequent adjustments and repairs. In fact, these klystrons cycling on one of the major sources of disturbances to the beam and drive the need for b feedback systems and adaptive disturbance cancelation.
is off-center, it is attracted to the wall. Because of these fields, known as wake bunch trajectory must be kept stable to better than 20 microns, the diameter of a h The first line of defense in trajectory control is passive: A set of 300 magn ing lenses (quadrupole electromagnets) spaced along the length of the accelerato beam like optical lenses focus light. Each lens gives a transverse kick to a parti tional to the distance the particle is from the correct trajectory. Hence if a bunc down the Linac slightly off-center, it will follow a roughly sinelike trajectory abo ter line as it speeds down the accelerator. The actual trajectory is a bit more comp is known as a betatron oscillation. A particle executes about 30 full oscillation down the Linac. In the second line of defense are quasi-DC dipole electromagnets. Near ea pole lens there are two dipole magnets: one which kicks the beam vertically and kicks it horizontaiiy. Operators can adjust these magnet strengths remotely via the trol system. They can be controlled individually by typing in the desired value also be controlled en masse by using an automated steering program which cal proper magnet settings to zero the beam orbit. Information for this program is obt 300 beam position monitors (BPMs) spaced down the Linac. This steering proced ically done several times a day. The beam position monitors are typically made of placed in a cylinder around the nominal beam trajectory. The beam is capacitive to these strips. If the beam is too high, it will be closer to the top strip and will ma pulse on that strip. Cables run from each strip to digitizing electronics. By com amplitude on the top and bottom strips (and similarly left and right), we determin position with a typical precision of 20 microns. In the third line of defense are steering feedback loops. Let’s consider the disturbances to the beam which make feedback necessary. 0
0
0
As mentioned above klystrons periodically break and hence are often being and off. This not only causes sudden changes in the beam energy but also c trajectory.
The electrons are accelerated near the crest of an electromagnetic wave. phase of the beam with respect to the wave determines the energy gain. Th difficult to keep stable over the 3-km length of the Linac and varies by a fe (which corresponds to a few trillionths of a second change in timing). The make the energy and trajectory change.
Operators purposely adjust some magnets to improve the beam quality. helps stabilize the beam downstream of these adjustments.
are placed one after the other along the Linac. A typical Linac steering loop (nu is illustrated in Fig. F.2. It measures and controls eight states: the position a the electron beam in both the horizontal and vertical directions and the same fo These eight states cannot be directly measured as there is no sensor which can termine the angle of the beam. Instead, the piece of the system labeled “transpo angles to BPM readings” takes the eight states (labeled “positions, angles at loo 40 beam position monitor (BPM) readings. These 40 readings correspond to the and vertical position for both electrons and positrons at each of the 10 BPM loc by loop n 1. The 40 measurements to determine eight states give a factor of 5 which helps reduce the effect of sensor noise and allows the system to keep ru when a BPM is broken. The boxes labeled “calculate positions, angles from ings” and “LQG feedback controller” represent our feedback system software the BPM readings to estimate the eight states which the controller then uses settings for the eight actuators that loop n 1 controls. These actuators consist o zontal dipole magnets and four vertical dipole magnets which bend the beam. Th junction above the “LQG feedback controller” box represents the effect of thes on the states of the beam. The signal flow lines in Fig. F.2 are labeled to indicate the number of sign The blocks are linear memoryless matrices, except for the z-’ I block which is pulse time delay and the “LQG Feedback Controller” block which contains a low in addition to its linear matrices. A lot of real-life complexity is left out of Fig. F.2 for clarity. All of the c of loop n 1 are spread out over a distance of 220 meters along the Linac. T and actuators are controlled by three different microprocessors (because of th tances and large number of devices, the SLC has a microprocessor for each 10 accelerator) which use simple point-to-point communication links to get all the point. This typical loop takes data and makes corrections at a rate of 20 Hz. It at the full accelerator pulse rate of 120 Hz due to speed limitations of the 16 M microprocessors on which the software currently runs. Each feedback loop is designed using our knowledge of the accelerator op state space formalism of control theory. The linear quadratic Gaussian (LQG used to design the optimum filters to minimize the RMS disturbance seen in the b there is a fair amount of white noise in the incoming beam disturbance, this filt over about six pulses. Hence the typical loop corrects most of a step change in As described so far, there is a problem with the system. We have seven loo in the Linac all looking at the same beam (see Fig. F. 1). Figure F.3 depicts the b tory in the region of two of these loops. Figure F.3(a) shows the trajectory on th after there has been a sudden disturbance (such as an operator adjusting a dip
+
+
+
states . 1
8
8 I
)
Transport from nton+l Accelerator
'I
8
Figure F.2
Disturbance between I nandn+l I loopn+I 8
4
I I
D
I I
----------------8
Diagram of typical feedback loop.
-
damage has been done: The loops have overshot the mark for a pulse. The p much worse with seven loops in a row. The overshoot can be reduced by havin respond more slowly but the system still overshoots and then rings for many system is stable and the ringing gradually dies out, but the overall response of not optimal; hence, the beam positions and angles have a larger RMS than need Disturbance location,
Loop
Loop +I
location
location 06-
EZ
I\
Trajectory fixed by loop
04-
a8 E -
0.2.
nE. b g
0
5%M - 0 2 . t
bE 1,000
1.200
1.400
1,600
1,800
-04
-0.6 .
Distance Along Line
Distance Along Line (m) (a)
\
2,000
(b)
Figure F.3 Feedback’s response to a disturbance. Shown is the beam trajectory on the first pulse pulse (b) after a sudden disturbance is introduced. The response of the two feedback loops shows t adaptive noise-canceling system.
The proper solution is to have each loop only correct for disturbances wh between it and the next upstream loop. This would completely eliminate the o caused by multiple loops correcting for the same disturbance.
F.5
ADDITION OF A MlMO ADAPTIVE NOISE CANCELER T O FAST FEEDBACK
+
An individual loop (say loop n 1) has only a few local beam position monito disturbances in the beam. It has no way to tell how far upstream the disturbanc Since we want loop n + 1 to correct for disturbancesdownstreamof loop n,but no the upstream disturbances can be thought of as noise. With this picture in m canceler can be used to solve our problem. For reasons explained in the nex must be an adaptive noise canceler. The rest of this section shows how an ada canceler can be used to solve this problem.
Actual
positi
to BP Calculate positions, angles to BPM readings
Y+
I
II I
I I
I I I I
I
' t 1
1
.
I I I
'?'
L J
8
LQ" feedt Controller
Err0r
\
8
Positions, angles at l o o pn+ I
7
1 I
Actual
I I I I
I I I 1
I
1
J
2
-
states
8
40
Transport positions, angles to BPM readings 1
". . _
Lillcuiate positions, angles
8
I I 1 I I I I I I
'
I
I I I
I I
I I I I
I I
I
to BPM readings
-----
*I MIMO adaptive noise canceller
reference Noise ' 7' ----------I I
Adaptive MIMO noise canceler added to the typical feedback loop.
Positions, angles at 'WPn
Figure F.4
F
To loop n+2
Positions, angles at loopn+ 1
at their desired setpoints (which are typically zero since we want the beam to go line down the center of the tube), as far as loop n 1 is concerned, these state Loop n reads some BPMs (beam position monitors) and calculates the positions from their readings. It both uses the numbers for its own feedback loop and sen a communications link (labeled “Measured positions, angles at loop n”) to loop uses them as its noise reference signal for its adaptive noise canceler. Similar information is carried to loop n I by the beam itself. Between th the beam is following its sinelike trajectory so that positions and angles transfor other. This is represented by the box labeled “Transport from n to n 1” and re accelerator dynamics between the two loops. It is very important to note that o is static; the transport of this beam pulse does not depend on the positions and a previous beam pulse. Hence, the box can be represented as a simple 8 x 8 matr in addition to this simple transport of the beam, there may be an additio bance between n and n + 1” which gets added in. This disturbance could be due t tripping off or an operator adjusting a magnet for example. It is this disturban n + 1 is intended to correct so it corresponds to the signal that we want the no to extract. The last box which needs explanation is the “LQG Feedback Controller.” sents the controller of feedback loop n I . As its input, the controller now take of the MIMO adaptive noise canceler which represents our best estimate of the “D between n and n 1 .” That is precisely what we want loop n + I to correct. Th the controller controls dipole magnets which steer the beam between n and n + output is shown summed into the positions and angles of the beam transported f In summary, before the implementation of the adaptive noise canceler, t seven feedback loops overcorrected for deviations in the position and angle o This happened because each feedback loop acted independently and hence they a correction for the same disturbance. The addition of the MIMO adaptive nois allows each loop to separate disturbances which happen immediately upstream which occur upstream of the previous loop. This cures the overcorrection prob This completes the description of the overall system. The next section ex adaptive algorithm used to update the weights.
+
+
+
+
+
F.6
ADAPTIVE CALCULATION
Before delving into the details of the adaptive calculation, it is worthwhile to ask tation is necessary at all. What is varying? The box labeled “Transport from n Fig. F.4 is what varies. It accounts for the sinelike trajectory, caused by the foc
real Linac. Note that the position and angle at the second loop are quite differen I percent error. It is this significant variation of the “Transportfrom n to n 1” w the use of an adaptive method for the noise canceler. The updates of the weights in the adaptive filter are done using the sequen sion (SER) algorithm. The equations used in the SER algorithm are explained in won’t give them here. Instead it is interesting to examine our experiences which use of this algorithm and to mention minor modifications that were made to ensu be robust. We started out using the least mean square (LMS) algorithm for updating t (matrix elements). This is the simplest algorithm, is very fast computationally,an successfully used in many applications. In the design phase of the project, simula done to check our algorithms. Two problems turned up.
+
0
As explained in [2], the LMS method is stable only if the learning rate is le
inverse of the largest eigenvalue of the input correlation matrix. For our pr the natural jitter of the beam due to magnet supply fluctuations and klystron that cause the variations of the positions and angles and hence the inform which the adaptation is done. The amplitude of this jitter can easily cha order of magnitude in a short time which in turn means the eigenvalues that amount. If the jitter increased too much, the LMS method would becom and our feedback system would malfunction, making things still worse. To doesn’t happen, we would have to pick a learning rate much smaller than optimum value which would result in a very slow convergence.
The LMS method has a different convergencerate for each eigenmode whic on the corresponding eigenvalue. Unless we carefully scaled our inputs, t gence of the slowest eigenmode would be much less than optimum.
To avoid these problems we went to the SER algorithm. Basically it adap mates the inverse of the input correlation matrix. This is used to scale the inputs the eigenvalues of the correlation matrix of the scaled inputs are equal to 1. Thi above two problems at the cost of added complexity and CPU time. Even with the SER method, the calculation of the weights becomes unstable time if the beam jitter suddenly increases. This is because it takes a while for its e the inverse of the input correlation matrix to converge to the new value. During th values for the weights have run away. This problem was found in simulation alon solution. We simply do not update the weights if the inverse correlation matrix is large updates.
F.7
EXPERIENCE ON THE REAL ACCELERATOR
The new communications links were installed on the accelerator while the so being written and tested. With the software ready, we first turned on just the gorithm. The results were not used to control the beam. After confirming that had converged to reasonable values, we turned on the use of the noise-cance As shown in Fig. F.5 the response to a step disturbance in the beam trajectory improved with the turn-on of the adaptive noise-canceling system. Disturbances
Disturbances
5.g
> a -0.5 (a)
.* -50
-40
-30
Time (sec.)
-0.4
-20
-70
-60
-50
-40
Time (sec.)
(b)
Figure F.5 Response of a chain of six feedback loops to a sudden disturbance in the incoming adaptive noise-canceling is off so there is a ringing caused by the overcorrection of many loops. noise-canceling is on so the whole chain of loops responds just like a single loop. In fact, the first work to correct the beam and the downstream loops dig virtually nothing.
Over the next few weeks we varied the learning rate to find the optimum would allow the adaptation to converge rapidly without having too much noise by the adaptive process. We settled on a learning rate of 0.001 and an adaptive of 10 Hz. This resulted in a convergence time of about 100 seconds. The system several days with learning rates of 0.1 and 0.0 1 and was completely stable, bu higher learning rates more random noise showed in the adaptive matrix elemen The adaptive noise-canceling addition to the fast feedback system has b stably in seven locations on the SLAC linear collider for over six months. Proba measure of its robustness and stability is that operators have made no middle phone calls for help to recover from a problem. In fact, there have been no signi lems with the system. Adaptive noise canceling has significantly improved the p of our feedback systems and helped us achieve our goal of accelerating two be
to the next. This shows a variation of over 30 degrees which is about 1 percent phase advance between the two loops. We have done many checks and convince that this is caused by a real variation in the focusing strengths in the Linac. A physicists are using plots like Fig. F.6 to try to track down the cause of these focusing strength so they can be fixed. This would make a still more stable beam 2450 2445
3
2440
&,
2435
a, Q)
3 2430 0
2 4 4 m
2 m
2425 2420 2415 2410 2405
1
2400‘ 0
1
0.5
1.5
I
Time (days) Figure F.6
The adaptively measured phase advance between two loops over a two-day period. T variation shows that the cascading of feedback would not have worked without the adaptive measu beam transport matrix.
ACKNOWLEDGEMENTS
The success of this project would not have been possible without the dedicate many people in the controls group at SLAC. Particularly important to the pr Stephanie Allison, Phyllis Grossberg, Linda Hendrickson, Dave Nelson, Rober Hamid Shoaee.
Bibliography for Appendix F
[l] F. ROUSE,S. CASTILLO, S. ALLISON, T. GROMME, L. HENDRICKSO K. KRAUTER, R. SASS,and H. SHOAEE, “Database driven fast feedback
Thirty Years of Adapti Neural Networks: Perceptro Madaline, an Backpropagatio
Fundamental developments in feedforward artificial neural networks from the past 30 are reviewed. The central theme of this paper is a description of the history, origin operating characteristics, and basic theory of several supervised neural network train gorithms including the Perceptron rule, the LMS algorithm, three Madaline rules, the backpropagation technique. These methods were developed independently, but w perspective of history they can all be related to each other. The concept which und these algorithms is the “minimal disturbance principle,” which suggests that during ing it is advisable to inject new information into a network in a manner which disturbs information to the smallest extent possible.
G.l
INTRODUCTION
This year marks the thirtieth anniversary of the Perceptron rule and the LMS algorithm early rules for training adaptive elements. Both algorithms were first published in In the years following these discoveries, many new techniques have been developed field of neural networks, and the discipline is growing rapidly. One early developme Steinbuch’s Learning Matrix [ 11, a pattern recognition machine based on linear discrim functions. In the same time frame, Widrow and his students devised Madame Rule I ( the earliest popular learning rule for neural networks with multiple adaptive elemen Other early work included the “mode-seeking’’ technique of Stark, Okajima, and W [3]. This was probably the first example of competitive learning in the literature, tho could be argued that earlier work by Rosenblatt on “spontaneous learning” [4,51 de I By Bernard Widrow and Michael A. Lehr.Reprinted, with permission, from Proceedings of rhe IEEE, No. 9 (September 1990), pp. 1415-1442.
also somewhat reminiscent of Albus’s adaptive CMAC, a distributed table-loo based on models of human memory [ 13, 141. Yet another approach related to bo is the associative reward-penaltyalgorithm of Barto and Anandan [ 151. This me associative reinforcement learning tasks, providing a link between pattern classi stochastic learning automata. In the 1970s Grossberg developed his adaptive resonance theory (ART of novel hypotheses about underlying principles which govern biological neu [16]. These ideas served as the basis for later work by Carpenter and Grossber three classes of ART architectures: ART 1 [ 171, ART 2 [18], and ART 3 [19] self-organizing neural implementations of pattern clustering algorithms. Othe theory on self-organizingsystems was pioneered by Kohonen with his work on fe [20,21]. In the early 1980s,Hopfield and others introducedouter product rules as we alent approaches based on the early work of Hebb [22] for training a class of rec nal feedback) networks now called Hopfield models [23, 241. More recently, tended some of the ideas of Hopfield and Grossberg to develop his adaptive b associative memory (BAM) [25], a network model employing differential as w bian and competitive learning laws. Another model utilizing a differential lear anism is Harry Klopf’s drive reinforcement theory [26], an extension of Hebbi which explains Pavlovian classical conditioning. Other significant models fro decade include probabilistic ones such as Hinton, Sejnowski, and Ackley’s Bolt chine [27, 281, which to oversimplify, is a Hopfield model that settles into sol simulated annealing process governed by Boltzmann statistics. The Boltzmann trained by a clever two-phase Hebbian-based technique. While these developments were taking place, adaptive systems research traveled an independent path. After devising their Madaline I rule, Widrow a dents developed uses for the Adaline and Madaline. Early applications includ others, speech and pattern recognition [29], weather forecasting [30], and adapt [31]. Work then switched to adaptive filtering and adaptive signal processing [3 tempts to develop learning rules for networks with multiple adaptive layers were ful. Adaptive signal processing proved to be a fruitful avenue for research with a involving adaptive antennas [33], adaptive inverse controls [34], adaptive nois [35], and seismic signal processing [32]. Outstanding work by R. W. Lucky an Bell Laboratories led to major commercial applicationsof adaptive filters and the rithm to adaptive equalization in high speed modems [36,37] and to adaptive ech for long distance telephone and satellite circuits [38]. After 20 years of resear
largely as a result of the clear framework within which they presented their ideas, succeeded in making it widely known. The elements used by Rumelhart and colleagues in the backpropagation ne fer from those used in the earlier Madaline architectures. The adaptive elements inal Madaline structure used hard-limiting quantizers (signums), while the elem backpropagation network use only differentiable nonlinearities, or “sigmoid” f ~ digital implementations, the hard-limiting quantizer is more easily computed t the differentiable nonlinearities used in backpropagation networks. In 1987, W Winter looked back at the original Madaline I algorithm with the goal of develo technique that could adapt multiple layers of adaptive elements which use the sim limiting quantizers. The result was Madaline Rule I1 [45]. David Andes of the U.S. Naval Weapons Center of China Lake, California Madaline I1 in 1988 by replacing the hard-limiting quantizers in the Adaline wi functions, thereby inventing Madaline Rule I11 (MRIII). Widrow and his student to recognize that this rule is mathematically equivalent to backpropagation. The outline above gives only a partial view of the discipline, and many lan coveries have not been mentioned. Needless to say, the field of neural networks becoming a vast one, and in one short survey we could not hope to cover the ject in any detail. Consequently, many significant developments, including som mentioned above, will not be discussed in this appendix. The algorithms describ limited primarily to those developed in our laboratory at Stanford and to related developed elsewhere, the most important of which is the backpropagation algo section headings indicate the range and coverage of this appendix:
G.1 Introduction G.2 Fundamental Concepts
G.3 Adaptation - The Minimal Disturbance Principle G.4 Error Correction Rules - Single Threshold Element
The a-LMS algorithm
*We should note, however, that in the field of variational calculus the idea of error backpropaga nonlinear systems existed centuries before Werbos first thought to apply this concept to neural netw past 25 years, these methods have been used widely in the field of optimal control, as discussed by L 3The term “sigmoid” is usually used in reference to monotonically increasing “S-shaped” function hyperbolic tangent. In this appendix, however, we generally use the term to denote any smooth nonlin at the output of a linear adaptive element. In other papers, these nonlinearities go by a variety of na “squashing functions,” “activation functions,” “transfer characteristics,” or ‘Wlreshold functions.”
The p L M S Algorithm Backpropagation for the Sigmoid Adaline Madaline Rule 111 for the Sigmoid Adaline
G.7 Steepest-Descent Rules - Multi-Element Networks Backpropagation for Networks Madaline Rule 111 for Networks
G.8 Summary
Information about the neural network paradigms not discussed in this p obtained from a number of other sources, such as the concise survey by Richard 1461, and the collection of classics by Anderson and Rosenfeld [47]. Much of the in the field from the 1960s is carefully reviewed in Nilsson’s monograph [48]. A of some of the more recent results is presented in Rumelhart and McClelland’s p volume set [49]. A paper by Moore [50] presents a clear discussion about ART of Grossberg’s terminology. Another resource is the DARPA Study report [Sl] w a very comprehensive and readable “snapshot”of the field in 1988.
G.2
FUdDAMENTAL CONCEPTS
Today we can build computers and other machines which perform a variety of w tasks with celerity and reliability unmatched by humans. No human can invert solve systems of differential equations at speeds which rival modem workstati theless, there are still many problems which have yet to be solved to our satisfac man-made machine but are easily disentangled by the perceptual or cognitive po mans, and often lower mammals, or even fish and insects. No computer vision rival the human ability to recognize visual images formed by objects of all shape tations under a wide range of conditions. Humans effortlessly recognize object environments and lighting conditions, even when obscured by dirt, or occlud objects. Likewise, the performance of current speech recognition technology compared to the performance of the human adult who easily recognizes words different people, at different rates, pitches, and volumes, even in the presence o or background noise. The problems solved more effectively by the brain than by the digital com ically have two characteristics: They are generally ill-defined, and they usually
processing high-resolution images requires a great deal of computation. Our bra plish this by utilizing massive parallelism, with millions and even billions of parts of the brain working together to solve complicated problems. Because soli erational amplifiers and logic gates can compute many orders of magnitude faste rent estimates of the effective computational speed of neurons in the brain, we be able to build relatively inexpensive machines with the ability to process as m mation as the human brain. This enormous processing power will do little to he problems, however, unless we can utilize it effectively. For instance, coordina thousands of processors which must efficiently cooperate to solve a problem is n task. If each processor must be programmed separately, and if all contingencies with various ambiguities must be designed into the software, even a relatively sim lem can quickly become unmanageable. The slow progress over the past 25 yea machine vision and other areas of artificial intelligence is testament to the difficu ciated with solving ambiguous and computationally intensive problems on von computers and related architectures. Thus, there is some reason to consider attacking certain problems by desig rally parallel computers which process information and learn by principles borro the nervous systems of biological creatures. This does not necessarily mean we tempt to copy the brain part for part. Although the bird served to inspire devel the airplane, birds do not have propellers, and airplanes do not operate by flap ered wings. The primary parallel between biological nervous systems and artifi networks is that each typically consists of a large number of simple elements tha are able to collectively solve complicated and ambiguous problems. Today, most artificial neural network research and application are accom simulating networks on serial computers. Speed limitations keep such networks small, but even with small networks some surprisingly difficult problems have led. Networks with fewer than 150 neural elements have been used successfully lar control simulations [52], speech generation [53,541,and undersea mine dete Small networks have also been used successfully in airport explosive detection [ systems [56, 571, and scores of other applications. Furthermore, efforts to deve lel neural network hardware are being met with some success, and such hardware available in the future for attacking more difficult problems like speech recognitio Whether implemented in parallel hardware or simulated on a computer, all works consist of a collection of simple elements that work together to solve pro basic building block of nearly all artificial neural networks, and most other ada tems, is the adaptive linear combiner.
input used to effect learning. The components of the input vector are weighted by efficients, the weight vector w/,= [ w ~, w,i l , w ~. . .~wnl, I T . The sum of the weig is then computed, producing a linear output, the inner product Sk = XcWk. The c of xk may be either continuous analog values or binary values. The weights are continuously variable and can take on negative as well as positive values.
Input Pattern Vector
-yJ
t
Error
wk
Weight Vector
Figure G.l
6
dk
Desired Response
Adaptive linear combiner.
During the training process, input patterns and corresponding desired re presented to the linear combiner. An adaptation algorithm automatically adjusts so that the output responses to the input patterns will be as close as possible spective desired responses. In signal processing applications, the most popular adapting the weights is the simple LMS (least mean square) algorithm 160,611, o the Widrow-Hoffdelta rule [MI. This algorithm minimizes the sum of squares o errors over the training set. The linear error 6k is defined to be the difference b desired response dk and the linear output S k , during presentation k. Having thi nal is necessary for adapting the weights. When the adaptive linear combiner is in a multi-element neural network, however, an error signal is often not directl for each individual linear combiner and more complicated procedures must be adapting the weight vectors. These procedures are the main focus of this appen
sgn(sk). The bias weight woL which is connected to a constant input, xo = +1, controls the threshold level of the quantizer. Input
______________________ x Ok= + I
Bias Input
-’’ I
Linear Output Binary output [+1,-1)
Desired Response Input (training signal) Figure C.2
An adaptive linear element (Adaline).
In single-element neural networks, an adaptive algorithm (such as the LMS or the Perceptron rule) is often used to adjust the weights of the Adaline so that correctly to as many patterns as possible in a training set which has binary desired Once the weights are adjusted, the responses of the trained element can be tested ing various input patterns. If the Adaline responds correctly with high probabili patterns that weie not included in the training set, it is said that generalization place. Learning and generalization are among the most useful attributes of Ad neural networks.
41n the neural network literature, such elements are often referred to as “adaptive neurons.” H conversation between David Hubel of Harvard Medical School and Bernard Widrow, Dr. Hubel po the Adaline differs from the biological neuron since it contains not only the neural cell body, but a synapses and a mechanism for training them.
old logic functions [62]. These are the set of logic functions that can be obtai possible weight variations. Figure G.3 shows a two-input Adaline element. Figure G.4 represents all nary inputs to this element with four large dots in pattern vector space. In th components of the input pattern vector lie along the coordinate axes. The A rates input patterns into two categories, depending on the values of the weight thresholding condition occurs when the linear outputs equals zero: s = x I w I +x2w2+wo=O,
therefore x2 =
WI
---XI
w2
Figure 6 . 3
-
wo -. w2
A two-input Adaline.
Figure G.4 graphs this linear relation, which comprises a separating line havin intercept given by WI
slope = --
w2
intercept =
wo U)2
The three weights determine slope, intercept, and the side of the separating line sponds to a positive output. The opposite side of the separating line correspond tive output. For Adalines with four weights, the separating boundary is a plane than four weights, the boundary is a hyperplane. Note that if the bias weight is ze arating hyperplane will be homogeneous - it will pass through the origin in pa
Figure G.4
Separating line in pattern space.
As sketched in Fig. (3.4, the binary input patterns are classified as follow (+l,+l) (+l,-l) (-1, -1) (--l,+l)
+I + +I + +1 +
+ -1.
This is an example of a linearly separable function. An example of a func is not linearly separable is the two-input exclusive NOR function: ( + 1 , + 1 ) + +1 ( + l , - l ) + -1 (-1, -1) + +1 ( - l , + l ) -+ -1.
No single straight line exists that can achieve this separation of the input patterns; out preprocessing, no single Adaline can implement the exclusive NOR functio With two inputs, a single Adaline can realize 14 of the 16 possible logic With many inputs, however, only a small fraction of all possible logic functions able, that is, linearly separable. Combinations of elements or networks of elem used to realize functions which are not linearly separable.
Capacity of Linear Classifiers
The number of training patterns or stimuli that an Adaline can learn to correctly an important issue. Each pattern and desired output combination represents an constraint on the weights. It is possible to have inconsistencies in sets of simul equalities just as with simultaneous equalities. When the inequalities (i.e., the.p
number of input patterns in the training set, and N,, the number of weights in t including the threshold weight, if used:
In Fig. G.5 this formula was used to plot a set of analytical curves which show the that a set of N p random patterns can be trained into an Adaline as a function N,/N,,,. Notice from these curves that as the number of weights increases, th pattern capacity of the Adaline C,T= 2 Nu, becomes an accurate estimate of the responses it can learn.
N , / N , , - Ratio of Input Patterns to Weights
Figure G.5
Probability that an Adaiine can separate a training pattern set as a function of the r
'Underlying theory for this result was discovered independently by a number of researchers inc others, R. 0.Winder [65],S. H. Cameron [66],and R. D. Joseph [67].
to the Adaline.6 If the input patterns to an Adaline are continuous valued an distributed (i.e., pattern positions are generated by a distribution function contai pulses), general position is assured. The general position assumption is often in pattern vectors are binary. Nonetheless, even when the points are not in gener the capacity results represent useful upper bounds. The capacity results apply to randomly selected training patterns. In mos of interest, the patterns in the training set are not random but exhibit some stati larities. These regularities are what make generalization possible. The number that an Adaline can leam in a practical problem often far exceeds its statistical c cause the Adaline is able to generalize within the training set and learns many of t patterns before they are even presented.
G.2.3
Nonlinear Classifiers
The linear classifier is limited in its capacity, and of course is limited to only linea ble forms of pattern discrimination. More sophisticatedclassifiers with higher ca nonlinear. Two types of nonlinear classifiers are described here. The first is a fix cessing network connected to a single adaptive element, and the other is the mu feedforward neural network.
Polynomial Discriminant Functions
Nonlinear functions of the inputs applied to the single Adaline can yield nonline boundaries. Useful nonlinearities include the polynomial functions. Consider illustrated in Fig. G.6 which contains only linear and quadratic input functions. T thresholding condition for this system is s = WO+XlWl
+x:wl,+xIx2w12+
XiW22 + x 2 w 2
= 0.
With proper choice of the weights, the separating boundary in pattern sp established as shown, for example, in Fig. G.7. This represents a solution for the NOR function of (G.5). Of course, all of the linearly separable functions are also The use of such nonlinearities can be generalized for more inputs than two and
6Patterns are in general position with respect to an Adaline with no threshold weight if any sub vectors containing no more than Nw members forms a linearly independent set, or equivalently, if no more input points in the N,-dimensional pattern space lie on a homogeneous hyperplane. For the m case involving an Adaline with a threshold weight, general position means that no set of N, or mo h e (Nu 1)-dimensional pattern space lie on a hyperplane not constrained to pass through the orig
-
Figure G.6
An Adaline with inputs mapped through nonlinearities.
degree polynomial functions of the inputs. Some of the first work in this area w D.F. Specht [68,69,70] at Stanford in the 1960s when he successfully applied discriminants to the classification and analysis of electrocardiographicsignals. W topic has also been done by Barron [7 1,72,73] and by A. G. Ivakhnenko [74] in Soviet Union.
t
x>
Ad~lr 1I I e
I I
0-
Adaline output = -1
Figure G.7
An elliptical separating boundary for realizing a function which is not linearly s
The polynomial approach offers great simplicity and beauty. Through it alize a wide variety of adaptive nonlinear discriminant functions by adapting o Adaline element. Several methods have been developed for training the poly criminant function. Specht developed a very efficient noniterative (i.e., single p the training set) training procedure, the polynomial discriminant method (PDM lows the polynomial discriminant function to implement a nonparametric clas
in a multidimensional Taylor series expansion of the desired response function. if appropriate trigonometric terms are used in place of the polynomial prepro Adaline’s weight solution will approximate the terms in the (truncated) multid Fourier series decomposition of a periodic version of the desired response fun choice of preprocessing functions determines how well a network will generali terns outside the training set. Determining “good” functions remains a focus of search 175,761. Experience seems to indicate that unless the nonlinearities are c care to suit the problem at hand, often better generalization can be obtained from with more than one adaptive layer. In fact, one can view multilayer networks as s networks with trainable preprocessors which are essentially self-optimizing. Madaline I
One of the earliest trainable layered neural networks with multiple adaptive ele the Madaline I structure of Widrow and Hoff [2, 771. Mathematical analyses of M were developed in the Ph.D. dissertations of Ridgway [78], Hoff [77], and Glanz [ early 196Os, a 1,000-weight Madaline I was built out of hardware [80] and used recognition research. The weights in this machine were memistors, electrically v sistors developed by Widrow and Hoff which are adjusted by electroplating a res
Madaline I was configured in the following way. Retinal inputs were co a layer of adaptive Adaline elements, the outputs of which were connected to a device that generated the system output. Methods for adapting such systems were at that time. An example of this kind of network is shown in Fig. G.8. Two Ad connected to an AND logic device to provide an output. With weights suitably chosen, the separating boundary in pattern space f tem of Fig. G.8 would be as shown in Fig. G.9. This separating boundary imple exclusive NOR function of (G.5). Madalines were constructed with many more inputs, with many more A ments in the first layer, and with various fixed logic devices such as AND, OR, an vote-taker elements in the second layer. Those three functions, illustrated in Fig all threshold logic functions. The given weight values will implement these three but the weight choices are not unique.
Feedforward Networks
The Madalines of the 1960s had adaptive first layers and fixed threshold funct second (output) layers [78,48]. The feedforward neural networks of today often h layers, and usually all layers are adaptive. The backpropagation networks of Rum
Figure G.8
A two-Adaline form of Madaline.
Separating
Figure G.9
Separating lines for Madaline of Fig. G.8
his colleagues [49] are perhaps the best-known examples of multilayer network connected three-layer’ feedforward adaptive network is illustrated in Fig. G . l l
’In Rumelhart and colleagues terminology, this would be called a four-layer network, following convention of counting layers of signals, including the input layer. For our purposes, we find it m count only layers of computing elements. We do not count as a layer the set of input terminal point
OR
Figure G.10
Fixed-weight Adaline implementations of AND, OR, and MAJ logic func
connected layered network, each Adaline receives inputs from every output in the layer. During training, the response of each output element in the network is com a corresponding desired response. Error signals associated with the output el readily computed, so adaptation of the output layer is straightforward. The fu difficulty associated with adapting a layered network lies in obtaining “error s hidden layer Adalines, that is, for Adalines in layers other than the output layer. propagation and Madaline 111 algorithms contain methods for establishing thes nals. There is no reason why a feedforward network must have the layered s Fig. G. 11. In Werbos’s developmentof the backpropagationalgorithm [39], in fa lines are ordered and each receives signals directly from each input component an
t
first-layer Adalines Figure C.11
f
second-layer Adalines
A three-layer adaptive neural network.
output of each preceding Adaline. Many other variations of the feedforward n possible. An interesting area of current research involves a generalized backp method which can be used to train “high order” or “sigma-pi” networks that in polynomial preprocessor for each Adaline 149, 821. One characteristic that is often desired in pattern recognition problems is of the network output to changes in the position and size of the input pattern or ious techniques have been used to achieve translation, rotation, scale, and time One method involves including in the training set several examples of each exem formed in size, angle, and position, but with a desired response which depends original exemplar [80]. Other research has dealt with various Fourier and Melli preprocessors [83, 841, as well as neural preprocessors [ 8 5 ] . Giles and colleagu veloped a clever averaging approach which removes unwanted dependencies fro nomial terms in high-order threshold logic units (polynomial discriminant fun and high-order neural networks [ 8 2 ] . Other approaches have considered Zernik [86], graph matching [87], spatially repeated feature detectors [9], and time-av puts [MI.
Capacity of Nonlinear Classifiers
An important consideration that should be addressed when comparing vario topologies concerns the amount of information they can store.* Of the nonlinea mentioned above, the pattern capacity of the Adaline driven by a fixed preproc posed of smooth nonlinearities is the simplest to determine. If the inputs to the smoothly distributed in position, the outputs of the preprocessing network will eral position with respect to the Adaline. Thus, the inputs to the Adaline will condition required in Cover’s Adaline capacity theory. Accordingly, the determ statistical pattern capacities of the system are essentially equal to those of the A
*We should emphasize that the information referred to here corresponds to the maximum num inputloutput mappings a network achieves with properly adjusted weights, not the number of bits o that can be stored directly into the network’s weights.
without preprocessingthe Adaline can separate only linearly separable sets, while line has no such limitation. A great deal of theoretical and experimental work has been directed toward ing the capacity of both Adalines and Hopfield networks [89, 90,91,92]. Som theoretical work has been focused on the pattern capacity of multilayer feedfo works, though some knowledge exists about the capacity of two-layer network sults are of particular interest because the two-layer network is surprisinglypow a sufficient number of hidden.elements, a signum network with two layers can any Boolean function.’ Equally impressive is the power of the two-layer sigmo Given a sufficient number of hidden Adaline elements, such networks can imp continuous input-output mapping to arbitrary accuracy [94, 95, 961. Although networks are quite powerful, it is likely that some problems can be solved more by networks with more than two layers. Nonfinite-orderpredicate mappings (e. nectedness problem [97]) can often be computed by small networks which use s back [98]. In the mid- I960s, Cover studied the capacity of a feedforward signum ne an arbitrary number of layersI0and a single output element [63,99]. He determin bound on the minimum number of weights N , needed to enable such a network any Boolean function defined over an arbitrary set of N p patterns in general po cently, Baum extended Cover’s result to multioutput networks and also used a co argument to find corresponding upper bounds for the special case of the two-la network [ 1001. Consider a two-layer fully connected feedforward network of sig lines which has N , input components (excluding the bias inputs) and Ny output co If this network is required to learn to map any set containing N p patterns which eral position to any set of binary desired response vectors (with N y components) from Baum’s results” that the minimum requisite number of weights N, can b by
From Eq. (G.8), it can be shown that for a two-layer feedforward network with se as many inputs and hidden elements as outputs (say, at least five times as many)
’This can be seen by noting that any Boolean function can be written in the sum of products form such an expression can be realized with a two-layer network by using the first-layer Adalines to imp gates, while using the second-layer Adalines to implement OR gates. “Actually, the network can be an arbitrary feedforward structure and need not be layered.
“The upper bound used here is Baum’s loose bound: min #hidden nodes 5 N1 rN,,/N,1 < N y ( N
where K, and K2 are positive numbers which are small terms if the network is few outputs relative to the number of inputs and hidden elements. It is easy to show that Eq.(G.8) also bounds the number of weights neede that N,, patterns can be learned with probability 1/2, except in this case the low N , becomes ( N y N p- 1)/(1+ l o g Z ( N p ) ) It. follows that Eq. (G.9) also serves t statistical capacity C, of a two-layer signum network. It is interesting to note that the capacity bounds (G.9) encompass the deter pacity for the single-layer network comprised of a bank of N , Adalines. In thi Adaline would have N u / N , weights, so the system would have a deterministic pacity of N , / N , . As N , becomes large, the statistical capacity also approac (for N , finite). Until further theory on feedforward network capacity is develop reasonable to use the capacity results from the single-layer network to estimate tilayer networks. Little is known about the number of binary patterns that layered sigmoi can leam to classify correctly. The pattern capacity of sigmoid networks canno than that of signum networks of equal size, however, because as the weights o network grow toward infinity, it becomes equivalent to a signum network wi vector in the same direction. Insight relating to the capabilities and operating p sigmoid networks can be winnowed from the literature [101, 102, 1031. A network’s capacity is of little utility unless it is accompanied by useful tions to patterns not presented during training. In fact, if generalization is not can simply store the associations in a lookup table and will have little need for a work. The relationship between generalizationand pattern capacity represents a tal trade-off in neural network applications: The Adaline’s inability to realize a is in a sense a strength rather than the fatal flaw envisioned by some critics of works [97], because it helps limit the capacity of the device and thereby improve to generalize. For good generalization, the training set should contain a number of patte several times larger than the network’s capacity (i.e., N , > z N , / N , ) . This ca stood intuitively by noting that if the number of degrees of freedom in a networ is larger than the number of constraints associated with the desired response fu N , N,,), the training procedure will be unable to completely constrain the weight work. Apparently, this allows effects of initial weight conditions to interfere w information and degrade the trained network’s ability to generalize. A detailed generalizationperformanceof signum networks as a function of training set size i in [104].
tention. NETtalk is a two-layer feedforward sigmoid network with 80 Adalines layer and 26 Adalines in the second layer. The network is trained to convert tex netically correct speech, a task well suited to neural implementation. The pronu most words follows general rules based upon spelling and word context, but ther exceptions and special cases. Rather than programming a system to respond each case, the network can learn the general rules and special cases by example One of the more remarkable characteristics of NETtalk is that it learns to words in stages suggestive of the learning process in children. When the output o is connected to a voice synthesizer, the system makes babbling noises during the e of the training process. As the network learns, it next conquers the general rule a child, tends to make a lot of errors by using these rules even when not appro the training continues, however, the network eventually abstracts the exceptions cases and is able to produce intelligible speech with few errors. The operation of NETtalk is surprisingly simple. Its input is a vector of sev ters (including spaces) from a transcript of text, and its output is phonetic inform responding to the pronunciation of the center (fourth) character in the seven-char field. The other six characters provide context which helps determine the desired To read text, the seven character window is scanned across a document in comp ory and the network generates a sequence of phonetic symbols which can be used a speech synthesizer. Each of the seven characters at the network’s input is a 29binary vector, with each component representing a different alphabetic characte tuation mark. A one is placed in the component associated with the represente while all other components are set to zero.’* The system’s 26 outputs correspond to 23 articulatory features and 3 add tures which encode stress and syllable boundaries. When training the network, response vector has zeros in all components except those which correspond to th features associated with the center character in the input field. In one experiment, and Rosenberg had the system scan a 1,024-wordtranscript of phonetically transc tinuous speech. With the presentation of each seven-character window, the system were trained by the backpropagation algorithm in response to the network’s ou After roughly 50 presentations of the entire training set, the network was able accurate speech from data the network had not been exposed to during training.
”The input representation often has a considerable impact on the success of a network. In NETta are sparsely coded in 29 components. One might consider instead choosing a 5-bit binary representa bit ASCII code. It should be. clear, however, that in this case the sparse representation helps simplify job of interpreting input characters as 29 distinct symbols. Usually the appropriate input encoding i to decide. When intuition fails, however, one sometimes must experiment with different encodings to works well.
those involving nonlinear control [52], system identification [52, 1061, signal [32], or decision making [571.
G.3
ADAPTATION
- T H E M I N I M A L DISTURBANCE PRIN
The iterative algorithms described in this appendix are all designed in accord gle underlying principle. These techniques - the two LMS algorithms, Mays’ the Perceptron procedure for training a single Adaline, the MRI rule for training Madaline, as well as MRII, MRIII, and backpropagation techniques for training Madalines - all rely upon the principle of minimal disturbance: Adapt to redu put error for the current training pattern, with minimal disturbance to respon learned. Unless this principle is practiced, it is difficult to simultaneously store t pattern responses. The minimal disturbance principle is intuitive. It was the mot that led to the discovery of the LMS algorithm and the Madaline rules. In fact, t gorithm had existed for several months as an error reduction rule before it was that the algorithm uses an instantaneous gradient to follow the path of steepest minimize the mean square error of the training set. It was then given the name “L mean square) algorithm.
ERROR CORRECTION RULES - SINGLE THRESHOL MENT
As adaptive algorithms evolved, principally two kinds of online rules have com One kind, error correcrion rules, alter the weights of a network to correct a cert tion of the error in the output response to the present input pattern. The other kin rules, alter the weights of a network during each pattern presentation by gradi with the objective of reducing mean square error, averaged over all training pat types of rules invoke similar training procedures. Because they are based upo objectives, however, they can have significantly different learning characteristi Error correction rules, of necessity, often tend to be ad hoc. They are mos when training objectives are not easily quantified, or when a problem does not le tractable analysis. A common application, for instance, concerns training neur that contain discontinuous functions. An exception is the a-LMS algorithm, a rection rule which has proven to be an extremely useful technique for finding s well-defined and tractable linear problems.
The a-LMS algorithm
The a-LMS algorithm or Widrow-Hoff delta rule applied to the adaptation of a s line (Fig. G.2) embodies the minimal disturbance principle. The weight update e the original form of the algorithm can be written as
The time index or adaptation cycle number is k. Wk+l is the next value of the we wk is the present value of the weight vector, and xk is the present input pattern v present linear error Ek is defined to be the difference between the desired respo the linear output Sk = w:xk before adaptation: Ek
A
= dk - wlxk.
Changing the weights yields a corresponding change in the error:
AEk = A(dk - wlxk) = - x : A w k .
In accordance with the a-LMS rule of Eq. (G. lo), the weight change is as follow AWk = Wk+l - Wk = a-
EkXk IXkl2
’
Combining Eqs. (G. 12) and (G. 13), we obtain
Therefore, the error is reduced by a factor of a as the weights are changed while h input pattern fixed. Presenting a new input pattern starts the next adaptation cycle error is then reduced by a factor of a,and the process continues. The initial weig usually chosen to be zero and is adapted until convergence. In nonstationary env the weights are generally adapted continually. The choice of a controls stability and speed of convergence [32]. For in vectors independent over time, stability is ensured for most practical purposes if O
Making a greater than 1 generally does not make sense, since the error would b rected. Total error correction comes with a = 1 . A practical range for a is 0.1 < a < 1.0.
I wk added to AWk, and AWk is parallel with th with Eq. (G.13), W ~ +equals tern vector &. From &. (G.12), the change in error is equal to the negative of Xk and Awn. Since the a-LMS algorithm selects AWk to be collinear with sired error correction is achieved with a weight change of the smallest possible When adapting to respond properly to a new input pattern, the responses to previ patterns are therefore minimally disturbed, on the average. The a-LMS algorithm corrects error, and if all input patterns are of equ minimizes mean square error [32]. The algorithm is best known for this proper
t" /-
X k = input pattern vector
-
= next weight vector
-
Awk=
weight vector change
wk = present weight vector
Figure C.12
G.4.2
Weight correction by the LMS rule.
Nonlinear Rules
The a-LMS algorithm is a linear rule which makes error corrections that are pro the error. It is known [ 1071 that in some cases this linear rule may fail to separ patterns that are linearly separable. Where this creates difficulties, nonlinear ru used. In the following paragraphs, we describe early nonlinear rules which.were Rosenblatt [ 108,5] and Mays[ 1071. These nonlinear rules also make weight vec collinear with the input pattern vector (the direction which causes minimal di changes which are based on the linear error but are not directly proportional to
Fixed Random Weights
X
Ft;$ik
1 Element
AnalogValued Retina Input Patterns
Output Decision
C
q
S arseRandom 6onnections Figure G.13
Z
t Desiredi Response v Threshold Elemente
Fixed Threshold Elements Rosenblatt’s a-Perceptron.
It is interesting to note that Rosenblatt’s Perceptron learning rule was firs in 1960 [ 1081, and Widrow and Hoff’s LMS rule was first presented the same y months later [61]. These rules were developed independently in 1959. The adaptive threshold element of the a-Perceptron is shown in Fig. G. 14 with the Perceptron rule makes use of the “quantizer error” F k , defined to be the between the desired response and the output of the quantizer % A Ek= d k
- yk.
The Perceptron rule, sometimes called the Perceptron convergence proce not adapt the weights if the output decision y k is correct, that is, if F k = 0. If decision disagrees with the binary desired response d k , however, adaptation is e adding the input vector to the weight vector when the error T k is positive, or subt input vector from the weight vector when the error z k is negative. Note that the error F k is always equal to either +2, -2, or 0. Thus, half the product of the input the quantizer error F k is added to the weight vector. The Perceptron rule is iden a-LMS algorithm, except that with the Perceptron rule, half of the quantizer er is used in place of the normalized linear error e k / l x k I* of the a-LMS rule. The
Binar outpu
(+1,-1
Desired Response Input (training signal) Figure G.14
The adaptive threshold element of the Perceptron
rule is nonlinear in contrast to the LMS rule which is linear (compare Figs. G.2 Nonetheless, the Perceptron rule can be written in a form which is very similar to rule of Eq. (G. 10):
Rosenblatt normally set a to one. In contrast to a-LMS, the choice of affect the stability of the Perceptron algorithm, and it affects convergence time initial weight vector is nonzero. Also, while a-LMS can be used with either anal desired responses, Rosenblatt’s rule can be used only with binary desired respo The Perceptron rule stops adapting when the training patterns are correctl There is no restraining force controlling the magnitude of the weights, however tion of the weight vector, not its magnitude, determines the decision function. T tron rule has been proven to be capable of separating any linearly separable se patterns [ 5 , 109,48, 1071. If the training patterns are not linearly separable, the algorithm goes on forever, and often does not yield a low-error solution, even if In most cases, if the training set is not separable, the weight vector tends to grav zerot3so that even if a is very small, each adaptation can dramatically affect th function implemented by the Perceptron.
I3This results because the length of the weight vector decreases with each adaptation that does linear output Sk to change sign and assume a magnitude greater than that before adaptation. Alth
presentation to be the smallest integer which corrects the output error in one pr If the training set is separable, this variant has all the characteristics of the fixed version with a set to 1, except that it usually reaches a solution in fewer presen
Mays’s Algorithms
In his Ph.D. dissertation, 11071, C. H. Mays described an “increment ada~tatio a “modified relaxation adaptation” rule. The fixed-incrementversion of the Perc is a special case of the increment adaptation rule. Increment adaptation in its general form involves the use of a “dead zone” ear output Sk, equal to f y about zero. All desired responses are f l (refer to Fig the linear output Sk falls outside the dead zone (Isk I 2 y ) , adaptation follows a variant of the fixed-increment Perceptron rule (with a / l X k (* used in place of a) ear output falls within the dead zone, whether or not the output response yk is c weights are adapted by the normalized variant of the Perceptron rule as though response yk had been incorrect. The weight update rule for Mays’s increment algorithm can be written mathematically as Wk+I =
1
wk+aG
&
wk +adk&
if/skl 2 y if (ski < y
where 2 k is the quantizer error of Eq. (G.17). With the dead zone y = 0, Mays’s increment adaptation algorithm reduce malized version of the Perceptron rule (G. 18). Mays proved that if the training p linearly separable, increment adaptation will always converge and separate the a finite number of steps. He also showed that use of the dead zone reduces se weight errors. If the training set is not linearly separable, Mays’s increment adap typically performs much better than the Perceptron rule because a sufficiently zone tends to cause the weight vector to adapt away from zero when any reaso solution exists. In such cases, the weight vector may sometimes appear to mea aimlessly, but it will typically remain in a region associated with relatively low a ror.
exceptions, for most problems this situation occurs only rarely if the weight vector is much longer th increment vector. I4The terms “fixed-increment’’and “absolute correction” are due to Nilsson [481Rosenblatt referr of these types, respectively, as quantized and nonquantized learning rules.
‘%e increment adaptation rule was proposed by others before Mays, though from a different pers
if W k f l
=
w k + f f t k $$ wk
€k=
0 and I S k ( 1 y
otherwise
where z k is the quantizer error of Eq. (G. 17). If the dead zone y is set to 00, this algorithm reduces to the a-LMS algori Mays showed that, for dead zone 0 < y < 1, and learning rate 0 < a 5 2, thi will converge and separate any linearly separable input set in a finite number of training set is not linearly separable, this algorithm performs much like Mays's adaptation rule. Mays's two algorithms achieve similar pattern separation results. The does not affect stability, although it does affect convergence time. The two ru their convergence properties, but there is no consensus on which is the better alg gorithms like these can be quite useful, and we feel that there are many more to and analyzed. The a-LMS algorithm, the Perceptron procedure, and Mays's algorithm used for adapting the single Adaline element or they can be incorporated into for adapting networks of such elements. Multilayer network adaption procedure some of these algorithms will be discussed below.
ERROR CORRECTION RULES - MULTI-ELEMENT NETWORKS
The algorithms discussed next are the Widrow-Hoff Madaline rule from the e now called Madaline Rule I (MRI) and Madaline Rule I1 (MRII), developed and Winter in 1987.
G.5.1
Madaline Rule I
The MRI rule allows the adaptation of a first layer of hard-limited (signum) A ments whose outputs provide inputs to a second layer, consisting of a single fixe logic element which may be, for example, the OR gate, AND gate, or Majority discussed previously. The weights of the Adalines are initially set to small rand Figure G.15 shows a Madaline I architecture with five fully connected first lines. The second layer is a majority element (MAJ). Because the second-laye ment is fixed and known, it is possible to determine which first-layer Adali
Command
Figure 6.15
A five-Adaline example of the Madaline I architecture.
adapted to correct an output error. The Adalines in the first layer assist each othe problems by automatic load-sharing. One procedure for training the network in Fig. G. 15 follows. A pattern is and if the output response of the majority element matches the desired response, tion takes place. However, if, for instance, the desired response is +1 and three Adalines read -1 for a given input pattern, one of the latter three must be ada 1 state. The element that is adapted by MRI is the one whose linear output S to zero, that is, the one whose analog response is closest to the desired response. the Adalines were originally in the - 1 state, enough of them are adapted to the make the majority decision equal 1. The elements adapted are those whose lin are closest to zero. A similar procedure is followed when the desired response is adapting a given element, the weight vector can be moved in the LMS direction to reverse the Adaline’s output (absolute correction or “fast” learning), or it can by the small increment determined by the a-LMS algorithm (statistical or “slow” The one desired response, dk, is used for all Adalines that are adapted. The pro also be modified to allow one of Mays’s rules to be used. In that event, for the ca considered (majority output element), adaptations take place if at least half of th either have outputs which differ from the desired response or have analog outputs in the dead zone. By setting the dead zone of Mays’s increment adaptation rule t weights can also be adapted by Rosenblatt’s Perceptron rule. Differencesin initial conditions and the results of subsequent adaptationcau ious elements to take “responsibility” for certain parts of the training problem. principle of load sharing is summarized thus: Assign responsibility to the Adafi lines that can most easily assume it.
+
+
to cycles of adaptation. These cycles would cause the weights of the entire Mad cle, preventing convergence. The adaptive system of Fig. G. 15 was suggested by common sense and w work well in simulations. Ridgway found that the probability that a given Ada adapted in response to an input pattern is greatest if that element had taken suc bility during the previous adapt cycle when the pattern was most recently pres division of responsibility stabilizes at the same time that the responses of ind ments stabilize to their share of the load. When the training problem is not per rable by this system, the adaptation process tends to minimize error probability is possible for the algorithm to “hang up” on local optima. The Madaline structure of Fig. G. 15 has two layers - the first layer consi tive logic elements, the second of fixed logic. There are a variety of fixed-lo that could be used for the second layer. A variety of MRI adaptation rules w by Hoff and described in his doctoral dissertation [77] which can be used with fixed-logic output elements. An easily described training procedure results wh put element is an OR gate. During training, if the desired output for a given in is 1, only the one Adaline whose linear output is closest to zero would be ada adaptation is needed, that is, if all Adalines give - 1 outputs. If the desired outp elements must give - 1 outputs, and any giving 1 outputs must be adapted. The MRI rule obeys the “minimal disturbance principle” in the following more Adaline elements are adapted than necessary to correct the output decisi dead-zone constraint. The elements whose linear outputs are nearest to zero because they require the smallest weight changes to reverse their output respo thermore, whenever an Adaline is adapted, the weights are changed in the dire input vector, providing the requisite error correction with minimal weight chan
+
+
G.5.2
Madaline Rule II
The MRI rule was recently extended to allow the adaptation of multilayer binar by Winter and Widrow with the introduction of Madaline Rule I1 (MRII) [45, 8 typical two-layer MRII network is shown in Fig. G .16. The weights in both laye tive. Training with the MRII rule is similar to training with the MRI algorithm. T are initially set to small random values. Training patterns are presented in a quence. If the network produces an error during a training presentation, we begi ing first-layer Adalines. By the “minimal disturbance principle,” we select the Adaline with the smallest linear output magnitude and perform a “trial adaptat verting its binary output. This can be done without adaptation by adding a pertu
Error)
. Desired Responses {+1,-1)
Figure G.16
Typical two-layer Madaline I1 architecture.
of suitable amplitude and polarity to the Adaline’s sum (refer to Fig. G. 16). If Hamming error is reduced by this bit inversion, that is, if the number of outp reduced, the perturbation As is removed and the weights of the selected Adal are changed by a-LMS in a direction collinear with the corresponding input ve direction which reinforces the bit reversal with minimal disturbance to the we versely, if the trial adaptation does not improve the network response, no weigh is performed. After finishing with the first element, we perturb and update other Adaline layer which have “sufficiently small” linear-output magnitudes. Further error red be achieved, if desired, by reversing pairs, triples, and so forth, up to some pre limit. After exhausting possibilities with the first layer, we move on to the nex proceed in a like manner. When the final layer is reached, each of the output adapted by a-LMS. At this point, a new training pattern is selected at random an dure is repeated. The goal is to reduce Hamming error with each presentation, the fully minimizing the average Hamming error over the training set. Like MRI, th can be modified so that adaptations follow an absolute correction rule or one of M rather than a-LMS. Like MRI, MRII can “hang up” on local optima.
G.6
STEEPEST-DESCENT RULES ELEMENT
- SINGLE THRESHOLD
Thus far, we have described a variety of adaptation rules that act to reduce a given of the error with the presentation of each training pattern. Often, the objective tion is to reduce error averaged in some way over the training set. The most co function is mean square error (MSE), although in some situations other error c
Adaptation of a network by steepest descent starts with an arbitrary initi for the system’s weight vector. The gradient of the mean square error function and the weight vector is altered in the direction corresponding to the negative sured gradient. This procedure is repeated, causing the MSE to be successively average and causing the weight vector to approach a locally optimal value. The method of steepest descent can be described by the relation
Wk+l = wk
+p ( - v k ) ,
where is a parameter that controls stability and rate of convergence, and vk of the gradient at a point on the MSE surface corresponding to W = wk. To begin, we derive rules for steepest descent minimization of the MSE with a single Adaline element. These rules are then generalized to apply to full-b networks. Like error correction rules, the most practical and efficient steepest d typically work with one pattern at a time. They minimize mean square error, app averaged over the entire set of training patterns.
G.6.1 Linear Rules
Steepest-descent rules for the single threshold element are said to be linear if wei are proportional to the linear error, the difference between the desired response linear output of the element, S k .
Mean Square Error Surface of the Linear Combiner
In this section we demonstrate that the MSE surface of the linear combiner of F quadratic function of the weights and is thus easily traversed by gradient desce Let the input pattern X k and the associated desired response dk be drawn fr tically stationary population. During adaptation, the weight vector varies so tha stationary inputs, the output Sk and error 6 k will generally be nonstationary. Ca taken in defining the mean square error since it is time varying. The only poss ensemble average, defined below. At the kth iteration, let the weight vector be wk. Squaring and expanding yields
X-vectorI6 then yields
PT
4 E [ d k X l ] = E[dk,dkXlk,...dkXnklT.
The input correlation matrix R is defined in terms of the ensemble average
R A E [xkxl]
This matrix is real, symmetric, and positive definite, or in rare cases, positive se The mean square error &. can thus be expressed as A
{k =
2
[Ek]~=~,, = E [d:] - 2PTWk 4-W l R W k .
Note that the mean square error is a quadratic function of the weights. It is a c perparaboloidal surface, a function that never goes negative. Figure G.17 show mean square error surface for a linear combiner with two weights. The position on the grid in this figure represents the value of the Adaline’s two weights. The he surface at each point represents the mean square error over the training set whe line’s weights are fixed at the values associated with the grid point. Adjusting t involves descending along this surface toward the unique minimum point (“the the bowl”) by the method of steepest descent. The gradient Vk of the mean square error function with W = Wk is obtain ferentiating Eq. (G.27):
This is a linear function of the weights. The optimal weight vector W * ,generally Wiener weight vector, is obtained from Eq. (G.28) by setting the gradient to zero
W* = R-’P. I6We assume here that X includes a bias component xok = + I .
Figure G.17
Typical mean square error surface of a linear combiner.
This is a matrix form of the Wiener-Hopf equation [ 120, 121, 1221. In the nex examine p-LMS, an algorithm which enables us to obtain an accurate estimate out first computing R-' and P.
The p-LMS Algorithm
The p-LMS algorithm works by performing approximate steepest descent o square error surface in weight space. Because it is a quadratic function of the w surface is convex and has a unique (global) minimum." An instantaneous gra upon the square of the instantaneous linear error is
LMS works by using this crude gradient estimate in place of the true gradien (G.28). Making this replacement into Eq. (G.21) yields
The instantaneous gradient is used because it is readily available from a single d The true gradient is generally difficult to obtain. Computing it would involv the instantaneous gradients associated with all patterns in the training set. Th impractical and almost always inefficient.
"Unless the autocorrelation matrix of the pattern vector set has rn zero eigenvalues. in which cas
MSE solution will be an m dimensional subspace in weight space [32].
Wk+l = Wk
+ 2pEkXk.
This is the p-LMS algorithm. The learning constant p determines stabilit vergence rate. For input patterns independent over time, convergence of the mea ance of the weight vector is ensured [32] for most practical purposes if 1
o < p < I -
tr[R] '
where tr[R] = C(diagona1 elements of R)is the average signal power of the X-v is, E ( X T X ) . With p set within this range," the p-LMS algorithm converges in to W', the optimal Wiener solution discussed above. A proof of this can be fou In the p-LMS algorithm, and other iterative steepest descent procedures, instantaneous gradient is perfectly justified if the step size is small. For small remain essentially constant over a relatively small number of training presentatio total weight change during this period will be proportional to
where denotes the mean square error function. Thus, on average the weights true gradient. It is shown in [32] that the instantaneous gradient is an unbiased e the true gradient.
Comparison of p-LMS and a-LMS
We have now presented two forms of the LMS algorithm, p-LMS (G.33)abo LMS ( G .10) in Section G.4.1. They are very similar algorithms, both using the L taneous gradient. a-LMS is self-normalizing, with the parameter (Y determinin tion of the instantaneous error to be corrected with each adaptation. p-LMS is a coefficient linear algorithm which is considerably easier to analyze than a-LMS. C
'*Horowitz and Senne [I231 have proven that Eq. (G.34) is not sufficient in general to guarantee of the weight vector's variance. For input patterns generated by a zero-meanGaussian process inde time, instability can occur in the worst case if p is greater than 1/(3tr[R]).
Replacing the error with its definition (G. 11) and rearranging terms yields
2 : -
We define a new training set of pattern vectors and desired responses, [ x k , d k ) , izing elements of the original training set as follows:2o
Equation (G.38) then becomes
This is the p-LMS rule of Eq. (G.33) with 2p replaced by a. The weight adap sen by the a-LMS rule are equivalent to those of the p-LMS algorithm prese different training set-the normalized training set defined by (G.39). The solut be reached by the p-LMSalgorithm is the Wiener solution of this training set,
w*= & - I
p,
is the input correlation matrix of the normalized training set and the vector
19Gradient noise is the difference between the gradient estimate and the true gradient. *‘The idea of a normalized training set was suggested by Demck Nguyen.
the learning constants a and p in the two algorithms. The stable range for a in algorithm given in Eq. (G. 15) can be computed from the correspondingrange for Eq. (G.34) by replacing R and p in Eq.(G.34) by R and a/2, respectively, and t . .?
that tr[R] is equal to one:
O
2
or
tr[RI o
G.6.2
Nonlinear Rules
The Adaline elements considered thus far use at their outputs either hard-limiting (signums), or no nonlinearity at all. The input-outputmapping of the hard-limitin is yk = sgn(sk). Other forms of nonlinearity have come into use in the past two de marily of the sigmoid type. These nonlinearitiesprovide saturation for decision m they have differentiable input-outputcharacteristics that facilitate adaptivity. We the definition of the Adaline element to include the possible use of a sigmoid in p signum, and then determine suitable adaptation algorithms. Figure G. 18 shows a “sigmoid Adaline” element which incorporates a sigm linearity. The input-output relation of the sigmoid can be denoted by Yk = sgm( ical sigmoid function is the hyperbolic tangent:
We shall adapt this Adaline with the objective of minimizing the mean square of t error z k , defined as rk
A
= dk - Yk = dk - Sgm(Sk).
Backpropagationfor the Sigmoid Adaline
Our objective is to minimize E[(E“k)2],averaged over the set of training patterns choice of the weight vector. To accomplish this, we shall derive a backpropag rithm for the sigmoid Adaline element. An instantaneous gradient is obtained input vector presentation, and the method of steepest descent is used to minimi was done with the p-LMSalgorithm of Eq.(G.33).
Error
Error ‘dk Desired Response
Figure 6.18
Adaline with sigmoidal nonlinearity.
Referring to Fig. G. 18, the instantaneous gradient estimate obtained durin tion of the kth input vector xk is given by
Differentiating Eq. (G.46) yields
We may note that
Therefore,
Substituting into Eq. (G.48) gives
Inserting this into Eq.(G.47) yields e k
= -2c‘&sgm’(Sk)Xk.
network, which will be studied below. Implementation of algorithm (G.54) is il Fig. G. 19. Input Pattern Vector
Weight Vector Wk
V
.-rl lk
x241 'nk
Figure G.19
Implementation of backpropagation for the sigmoid Adaline element.
If the sigmoid is chosen to be the hyperbolic tangent function (G.45), then tive sgm'(sk) is given by
Madaline Rule 111 for the Sigmoid Adaline
The implementation of algorithm (G.54), illustrated in Fig. G. 19, requires accur tion of the sigmoid function and its derivative function. These functions may not accurately when implemented with analog hardware. Indeed, in an analog net Adaline will have its own individual nonlinearities. Difficulties in adaptation ha countered in practice with the backpropagation algorithm because of imperfec nonlinear functions.
Perturbation
Figure C.20
Implementation of the MRIII algorithm for the sigmoid Adaline eleme
An instantaneous estimated gradient can be obtained as follows:
Since A s is small,
Another way to obtain an approximate instantaneous gradient by measu fects of the perturbation As can be obtained from Eq. (G.57).
Accordingly, there are two forms of the MRIII algorithm for the sigmoid Adalin based on the method of steepest descent, using the estimated instantaneous grad
of implementational convenience. The algorithm of Eq. (G.60) is illustrated in Regarding algorithm (G.61), some changes can be made to establish a poi est. Note that, in accord with Eq. (G.46),
Adding the perturbation A s causes a change in ck equal to
Now, Eq. (G.61) may be rewritten as
Since A s is small, the ratio of increments may be replaced by a ratio of different giving Wk+l
2
- aYk wk + 2pEk-Xk
= Wk
ask
+ 2pgksgrn'(sI;)xk.
This is identical to the backpropagation algorithm (G.54) for the sigmoi Thus, backpropagation and MRIII are mathematically equivalent if the perturba small, but MRIII is robust, even with analog implementations.
MSE Surfaces of the Adaline
Figure G.2 1 shclws a linear combiner connected to both sigmoid and signum devi errors, E , Zk, and E" are designated in this figure. They are linear error = E = d - s sigmoid error = E" = d - sgm(s)
=
signum error = E =d - sgn(sgm(s)) = d - sgn(s)
To demonstrate the nature of the square error surfaces associated with t types of error, a simple experiment with a two-input Adaline was performed. Th was driven by a typical set of input patterns and their associated binary (+1, responses. The sigmoid function used was the hyperbolic tangent. The weights c % been adapted to minimize the mean square error of e,?, or E . The mean square faces of E[(€)*],E[(?)*],E[(;)*] plotted as functions of the two weight values, in Figs. (3.22, G.23, and G.24, respectively.
Desired Response Figure G.21
The linear, sigiiioid. and \ignurn errors of the Adaline.
Figure G.22
Example MSE \urface of linear error.
In nonlinear neural networks, gradient methods generally work better w rather than signum nonlinearities. Smooth nonlinearities are required by the backpropagation techniques. Moreover, sigmoid networks are capable of form representations which are more complex than simple binary codes, and thus, the can often form decision regions which are more sophisticated than those asso similar signum networks. In fact, if a noiseless infinite-precision sigmoid Adali
Figure G.23
Example MSE surface of sigmoid error.
Figure G.24
Example MSE surface of signum error.
constructed, it would be able to convey an infinite amount of information at each This is in contrast to the maximum Shannon information capacity of one bit asso each binary element. The signum does have some advantages over the sigmoid in that it is easie ment in hardware and much simpler to compute on a digital computer. Furthe outputs of signums are binary signals which can be efficiently manipulated by d puters. In a signum network with binary inputs, for instance, the output of each l biner can be computed without performing weight multiplications. This invol adding together the values of weights with 1 inputs and subtracting from this of all weights that are connected to - 1 inputs. Sometimes a signum is used in an Adaline to produce decisive output deci error probability is then proportional to the mean square of the output error E" mize this error probability approximately, one can easily minimize E[ ( E ) ~ ]ins rectly minimizing E[(z)z][60].However, with only a little more computation
+
Figure 6.25
Example two-layer backpropagation network architecture.
We now study rules for steepest descent minimization of the MSE associated networks of sigmoid Adaline elements. Like their single-element counterpart practical and efficient steepest descent rules for multi-element networks typically one pattern presentation at a time. We will describe two steepest descent rules element sigmoid networks: backpropagation and Madaline Rule 111.
G.7.1
Backpropagation for Networks
The publication of the backpropagation technique by Rumelhart and colleagu unquestionablybeen the most influential development in the field of neural netwo the past decade. In retrospect, the technique seems simple. Nonetheless, large early neural network research dealt almost exclusively with hard-limiting nonline idea never occurred to neural network researchers throughout the 1960s. The basic concepts of backpropagation are easily grasped. Unfortunately, ple ideas are often obscured by relatively intricate notation, so formal derivat backpropagation rule are often tedious. We present an informal derivation of the and illustrate how it works for the simple network shown in Fig. G.25. A m derivation would require the use of ordered derivatives to make precise which w treated as variables in each of the partial derivatives used below [ 1261. The backpropagation technique is a nontrivial generalization of the sing Adaline case of Section G.6.2. When applied to multi-element networks, the bac tion technique adjusts the weights in the direction opposite the instantaneous erro
Now, however, Wk is a long n-component vector of all weights in the entire net instantaneous sum squared error E: is the sum of the squares of the errors at each outputs of the network. Thus,
In the network example shown in Fig. G.25, the sum square error is given by E2
= (dl - Y d 2
+ (dz - Y 2 l 2 ,
dom numbers. The algorithm will not work properly with multilayer networks weights are either zero or poorly chosen nonzero values.2'. We can get some idea about what is involved in the calculations associat backpropagation algorithm by examining the network of Fig. (3.25. Each of th circles represents a linear combiner, as well as some associated signal paths for propagation, and the corresponding adaptive machinery for updating the weigh tail is shown in Fig. (3.26. The solid lines in these diagrams represent forward s through the network, and the dotted lines represent the separate backward paths t in association with calculations of the square error derivatives 8 . From Fig. G.25, the calculations associated with the backward sweep are of a complexity which equal to that represented by the forward pass through the network. The backw requires the same number of function calculations as the forward sweep, but no w tiplications in the first layer. As stated above, after a pattern has been presented to the network, and th error of each output has been calculated, the next step of the backpropagatio involves finding the instantaneous square error derivative S associated with eac junction in the network. The square error derivative associated with the j th Ada C is defined asZ2
Each of these derivatives in essence tells us how sensitive the sum square out the network is to changes in the linear output of the associated Adaline element The instantaneous square error derivatives are first computed for each ele output layer. The calculation is simple. As an example, below we derive the r
"Recently, Nguyen has discovered that a more sophisticated choice of initial weight values in hid lead to reduced problems with local optima and dramatic increases in network training speed [ 1021. evidence suggests that it is advisable to choose the initial weights of each hidden layer in a quasi-ra which ensures that at each position in a layer's input space the outputs of all but a few of its Ad saturated, while ensuring that each Adaline in the layer is unsaturated in some region of its input spa method is used, the weights in the output layer are set to small random values.
'*In Fig. G.25, all notation follows the convention that superscripts within parentheses indicate the of the associated Adaline or input node. while subscripts identify the associated Adaline(s) within a
Figure G.26
Detail of linear combiner and associated circuitry in backpropagation netw
the derivative associated with the top Adaline element in the ou pression for from Eq. (G.71), of Fig. G.25. We begin with the definition of
Expanding the squared error term e2 by Eq. (G.70) yields
We note that the second term is zero. Accordingly,
6;” = c/”sgm’ (sI( 2 )).
Note that this corresponds to the computation of 6;” as ill! strated in Fig. (3.25 of S associated with the other output element in the figure can be expressed in a fashion. Thus each square error derivative 6 in the output layer is computed by the output error associated with that element by the derivative of the associate nonlinearity. Note from Eq. (G.55)that if the sigmoid function is the hyperbo Eq. ((3.78) becomes simply
Si”
=
q ’ ( 1 - (y1)2).
Developing expressions for the square error derivatives associated with h is not much more difficult (refer to Fig. G.25). We need an expression for S!’ error derivative associated with the top element in the first layer of Fig. G.25. Th 8:” is defined by
Expanding this by the chain rule, noting that E~ is determined entirely by the v and yields
sr),
Using the definitions of Si2’ and Sf’, and then substituting expanded versions linear outputs s!*’ and si2)gives
Referring to Fig. (3.25, we can trace through the circuit to verify that 6:” is in accord with Eqs. (G.86) and (G.87). The easiest way to find values of S for all t elements in the network is to follow the schematic diagram of Fig. (3.25. Thus, the procedure for finding S(e), the square error derivative associated w Adaline in hidden layer l , involves respectively multiplying each derivative See ated with each element in the layer immediately downstream from a given Ada weight which connects it to the given Adaline. These weighted square error deri then added together, producing an error term c(‘), which, in turn, is multiplied by the derivative of the given Adaline’s sigmoid function at its current operating poin work has more than two layers, this process of backpropagating the instantaneous ror derivatives from one layer to the immediately preceding layer is successivel until a square error derivative S is computed for each Adaline in the network. Th shown at each layer by repeating the chain rule argument associated with Eq.(G We now have a general method for finding a derivative S for each Adaline the network. The next step is to use these 8’s to obtain the corresponding gradi sider an Adaline somewhere in the network which, during presentation k, has a w tor wk, an input vector xk, and a linear output .sk = WIT&. The instantaneous gradient for this Adaline element is
This can be written as
Note that wk and xk are independent so
Therefore,
For this element,
Thus, after backpropagating all square error derivatives, we complete a back iteration by adding to each weight vector the corresponding input vector scale sociated square error derivative. Equation (G.94) and the means for finding & c general weight update rule of the backpropagation algorithm. There is a great similarity between Eq. (G.94) and the p-LMS algorithm one should view this similarity with caution. The quantity 6 k . defined as a sq derivative, might appear to play the same role in backpropagation as that played in the p-LMS algorithm. However, Sk is not an error. Adaptation of the given effected to reduce the squared output error E:, not 8 k of the given Adaline or o Adaline in the network. The objective is not to reduce the 8 k ’ S of the network, b E: at the network output. It is interesting to examine the weight updates that backpropagation imp Adaline elements in the output layer. Substituting Eq. (G.77) into Eq. (G.94) the Adaline which provides output yl in Fig. G.25 is updated by the rule
Wk+l = wk
+ 2pg,‘2’sgm’(s:2))Xk.
This rule turns out to be identical to the single Adaline version (G.54) of the ba tion rule. This is not surprising since the output Adaline is provided with both in and desired responses, so its training circumstance is the same as that experie Adaline trained in isolation. There are many variants of the backpropagation algorithm. Sometimes, t is reduced during training to diminish the effects of gradient noise in the weigh extension is the momentum technique [44] which involves including in the we vector AWk of each Adaline a term proportional to the corresponding weight c the previous iteration. That is, Eq. (G.94) is replaced by a pair of equations
where the momentum constant 0 Iq < 1 is in practice usually set to something or 0.9. The momentum technique low-pass filters the weight updates and there resist erratic weight changes due either to gradient noise or to high spatial freque mean square error surface. The factor (1 - q ) in Eq. (G.96) is included to give the gain of unity so that the learning rate 1-1 does not need to be stepped down as the constant q is increased. A momentum term can also be added to the update e other algorithms discussed in this appendix. A detailed analysis of stability issue with momentum updating for the p-LMS algorithm, for instance, has been d Shynk and Roy [ 1271.
where again 0 < q < 1. The weight updates determined by Eqs. ((3.98) and (G.9 a network find a solution when a relatively flat local region in the mean square
face is encountered. The weights move by the same amount whether the surfac inclined. It is reminiscent of a-LMS because the gradient term in the weight up tion is normalized by a time-varying factor. The weight update rule could be fu ified by including terms from both techniques associated with Eqs. (G.96) throu Other methods for speeding up backpropagation training include Fahlman’s popu prop method [ 1281, as well as the delta-bar-delta approach reported in an excellen Jacobs [ 1 29].24 One of the most promising new areas of neural network research involves agation variants for training various recurrent (signal feedback) networks. Rece propagation rules have been devised for training recurrent networks to learn sta ations [ 130, 1311. More interesting is the online technique of Williams and Zi which allows a wide class of recurrent networks to learn dynamic associations tories. A more general and computationally viable variant of this technique ha vanced by Narendra and Parthaswathy [ 106). These online methods are general a well-known steepest descent algorithm for training linear IIR filters [ 133, 321. An equivalent technique which is usually far less computationally intensiv suited for offline computation also exists [39,44, 1341. This approach, called “bac tion through time,” has been used by Nguyen and Widrow [52] to enable a neur to learn without a teacher how to back up a computer-simulated trailer truck to dock (Fig. G.27). This is a complicated and highly nonlinear steering task. Ne with just six inputs providing information about the current position of the tru layer neural network with only 26 Adalines was able to learn of its own accord to problem. Once trained, the network could successfully back up the truck from position and orientation in front of the loading dock. ”“Clean” gradient estimates are those with little gradient noise
24Jacobs’s paper, like many other papers in the literature, assumes for analysis that the true gra than instantaneous gradients are used to update the weights, that is, that weights are changed perio after all training patterns are presented. This eliminates gradient noise but can slow down training e the training set is large. The delta-bar-delta procedure in Jacobs’s paper involves monitoring chang gradients in response to weight changes, It should be possible to avoid the expense of computing the explicitly in this case by instead monitoring changes in the outputs of, say, two momentum filters w time constants.
I
Final State
Figure G.27
G.7.2
Example truck backup sequence.
Madaline Rule 111 for Networks
It is difficult to build neural networks with analog hardware which can be trained by the popular backpropagationtechnique. Attempts to overcome this difficulty the development of the MRIII algorithm. A commercial analog neurocomputin primarily on this algorithm has already been devised [ 1351. The method descr section is a generalizationof the single Adaline MRIII technique (G.60). The mu generalization of the other single element MRIII rule (G.61) is described in [13 The MRIII algorithm can be readily described by referring to Fig. G.28 this figure shows a simple two-layer feedforward architecture, the procedure t oped will work for neural networks with any number of Adaline elements in a ward structure. In [ 1361, we discuss variants of the basic MRIII approach that all descent training to be applied to more general network topologies, even those feedback. Assume that an input pattern X and its associated desired output response are presented to the network of Fig. G.28. At this point, we measure the sum squ
Desired -Responses
Figure G.28
Example two-layer Madaline 111 architecture.
+
response error E’ = (dl - y1)* ( d ~ y ~ ) ’= 6: + E ; . We then add a small quan selected Adaline in the network, providing a perturbation to the element’s linear perturbation propagates through the network, and causes a change in the sum of An easily measured ratio is of the errors, A(&’) = A(€:
+ €2).
A (E’) - -A As
(6:
+ €22)
As
--a -
(E’)
as
Below we use this to obtain the instantaneous gradient of E: with respect to the we of the selected Adaline. For the kth presentation, the instantaneous gradient is
Replacing the derivative with a ratio of differences yields
The idea of obtaining a derivative by perturbing the linear output of the sel line element is the same as that expressed for the single element in Section G.6 that here the error is obtained from the output of a multi-element network rather the output of a single element. The gradient (G.102) can be used to optimize the weight vector in accor method of steepest descent:
Maintaining the same input pattern, one could either perturb all the elem network in sequence, adapting after each gradient calculation, or else the deriva
because each weight update changes the associated unperturbed error.
G.7.3
Comparison of MRlll with MRll
MRIII was derived from MRII by replacing the signum nonlinearities with sig similarity of these algorithms becomes evident when comparing Fig. G.28, r MRIII, with Fig. G.16, representing MRII. The MRII network is highly discontinuous and nonlinear. Using an instant dient to adjust the weights is not possible. In fact, from the MSE surface for Adaline that we presented in Sectiun (3.6.2,it is clear that even gradient descen which use the true gradient could run into severe problems with local minima. adding a perturbation to the linear sum of a selected Adaline element is workabl If the Hamming error has been reduced by the perturbation, the Adaline is ad verse its output decision. This weight change is in the LMS direction, along i If adapting the Adaline would not reduce network output error, it is not adap in accord with the minimal disturbance principle. The Adalines selected for po tation are those whose analog sums are closest to zero, that is, the Adalines w adapted to give opposite responses with the smallest weight changes. It is useful with binary f1 desired responses, the Hamming error is equal to 1/4 the sum s Minimizing the output Hamming error is therefore equivalent to minimizing the square error. The MRIII algorithm works in a similar manner. All the Adalines in the work are adapted, but those whose analog sums are closest to zero will usually most strongly, because the sigmoid has its maximum slope at zero, contributing dient values. As with MRII, the objective is to change the weights for the give sentation to reduce the sum square error at the network output. In accord with t disturbance principle, the weight vectors of the Adaline elements are adapted direction, along their X-vectors, and are adapted in proportion to their capabil ducing the sum square error (the square of the Euclidean error) at the output.
G.7.4
Comparison of MRlll with Backpropagation
In Section G.6.2, we argued that, for the sigmoid Adaline element, the MRII (G.61) is essentially equivalent to the backpropagation algorithm ((3.54). The ment can be extended to the network of Adaline elements, demonstrating that if and adaptation is applied to all elements in the network at once, then MRIII is equivalent to backpropagation. That is, to the extent that the sample derivati
gation. Steepest-descent training of multi-layer networks implemented by compu tion or by precise parallel digital hardware is usually best carried out by backpr During each training presentation, the backpropagation method requires only o computation through the network followed by one backward computation in ord all the weights of an entire network. To accomplish the same effect with the form that updates all weights at once, one measures the unperturbed error followed ber of perturbed error measurements equal to the number of elements in the netw process can represent a significant amount of computation. If a network is to be implemented in analog hardware, however, experience that MRllI offers strong advantages over backpropagation. Comparison of Fig. Fig. G.28 demonstrates the relative simplicity of MRIII: All the apparatus for propagation of error-related signals is eliminated, and the weights do not need to nals in both directions (see Fig. (3.26). MRIII is a much simpler algorithm to b understand, and in principle it produces the same instantaneous gradient as the b gation algorithm. The momentum technique and most other common variants o propagation algorithm can be applied to MRIII training.
G.7.5
MSE Surfaces of Neural Networks
In Section (3.6.2, “typical” mean square error surfaces of sigmoid and signum Ada shown indicating that sigmoid Adalines are much more conducive to gradient a than signum Adalines. The same phenomena result when Adalines are incorpo multi-element networks. The MSE surfaces of MRII networks are reasonably c will not be explored here. In this section we examine only MSE surfaces from backpropagation training problem with a sigmoidal neural network. In a network with more than two weights, the MSE surface is high-dimen difficult to visualize. It is possible, however, to look at slices of this surface b the mean square error surface created by varying two of the weights while holdi ers constant. The surfaces plotted in Figs. G.29 and G.30 show two such slices o surface from a typical learning problem involving, respectively, an untrained sigm work and a trained one. The first surface resulted from varying two first-layer an untrained network. The second surface resulted from varying the same two we the network was fully trained. The two surfaces are similar, but the second one ha minimum which was carved out by the backpropagation learning process. Figure (3.32 resulted from varying a different set of two weights in the same network. is the result from varying a first-layer weight and third-layer weight in the untr
Figure G.29
Example MSE surface of an untrained sigmoidal network as a function of two first
Figure (3.30 Example MSE surface of a trained siginoidal network as a function of two first-l
work while Fig. G.32 is the surface that resulted from varying the same two w the netwcrk was trained. By studying many plots, it becomes clear that backpropagation and MRIII ject to convergence on local optima. The same is true for MRII. The most comm for this is the sporadic addition of noise to the weights or gradients. Some of the annealing” methods [ 1371 do this. Another method involves retraining the netw times using different random initial weight values until a satisfactory solution i Solutions found by people in everyday life are usually not optimal, but m are useful. If a local optimum yields satisfactory performance, often there is sim to search for a better solution.
SUMMARY
This year (1990) is the thirtieth anniversary of the publication of the Percept Rosenblatt and the LMS algorithm by Widrow and Hoff. It has also been I6 year bos first published the backpropagation algorithm. These learning rules and se
Figure G.31 Example MSE surface of an untrained sigrnoidal network as a function of a first-lay a third-layer weight.
Figure G.32 Example MSE surface of a trained sigrnoidal network as a function of a first-layer third-layer weight.
have been studied and compared. Although they differ significantly from each all belong to the same “family.” A distinction was drawn between error correction rules and steepest des The former includes the Perceptron rule, Mays’s rules, the a-LMS algorithm, t Madaline I rule of 1962, and the Madaline I1 rule.’The latter includes the I-LMS the Madaline I11 rule, and the backpropagation algorithm. The chart in Fig. G.33 c the learning rules that have been studied. Although these algorithms have been presented as established learning should not gain the impression that they are perfect and frozen for all time. Var possible for every one of them. They should be regarded as substrates upon whi
f ‘3‘“‘h
Nonlinear 6JnW
MRI MRII
Figure G.33
Nonlinear
hear
Perceptron Mays
a-LMS
Learning rules.
new and better rules. There is a tremendous amount of invention waiting “in We look forward to the next 30 years.
Acknowledgments
This work was sponsored by SDIO Innovative Science and Technology Office an by ONR under contract #NO001 4-86-K-07 18, by the Department of the Army B & E Center under contracts #DAAK70-87-P-3 134 and #DAAK70-89-K-0001 from the Lockheed Missiles and Space Company, by NASA under contract # and by Rome Air Development Center under contract #F30602-88-D-0025, sub 2 l -T22-S l .
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Neural Control System
A variety of methods for control for nonlinear systems were presented in Chapter 1 1 appendix describes additional methods for nonlinear control which are based on neur works. The networks of interest are multilayered, and adaptation is done with the bac agation algorithm. For a description of these networks and the backpropagation le algorithm, refer to Appendix G.
H.l
A NONLINEAR ADAPTIVE FILTER BASED O N NEURAL NETWORKS
Figure H. 1 shows the architecture of a nonlinear filter made by connecting a multilaye ral network to the taps of a tapped delay line. The connections in the network are va weights, as indicated in the figure. Each of the neural elements contains a sigmoida linearity. The error at the filter output is the difference between the desired response a network response. The error is used by the backpropagationalgorithm to adapt the w The same filter of Fig. H.1 can be represented more simply by the diagram of Fi Furthermore, with even less detail, a nonlinear adaptive filter can be drawn in a gener as in Fig. H.3. It is understood that the filter is of the form of a tapped delay line with a layered neural network attached. The error signal is used with backpropagationto ad weights. The number of weights, the number of neurons, and the number of layers need to be specified in a given case.
H.2
A MIMO NONLINEAR ADAPTIVE FILTER
A MIMO filter is shown in Fig. H.4.Once again, this is a transversal (tapped delay filter with neural networks (connected to the delay line taps). The weights are adap backpropagation using both error signals. This MIMO filter can be represented by th gram of Fig. H.3.As such, the input, output, error, and desired response signals wo vector rather than scalar.
+
Sigmoid
F Desired response d
* Figure H.l
An adaptive nonlinear filter composed of a tapped delay line and a three-layer n
Summers and + sigmoids
I Figure H.2
I
b
A simpler diagram for the adaptive filter of Fig. H. 1
o
Input
Nonlinear adaptive filter
Desired response d Figure H.3
A general representation for an adaptive transversal filter based on neural netw
Error # 1 1 Input 4 #I
z-
Z-
Input #2
I
4z I
-e-
Error €2 # 2
d2
Desir respon Figure H.4
A two-input two-output MIMO adaptive filter.
compared with the desired response, and the error is used to adapt B ( z ) with co LMS. A ( z ) is adapted using the same error filtered by B - ' ( z ) . This works well B ( z ) is adapted slowly and is always minimum-phase.
Input
d Desired response
Figure H.5
Training a cascade of adaptive filters by means of the filtered-€ LMS algori
If B ( z ) is nonminimum-phase, then a good inverse would be a delayed in sume that an offline process is used to get B ~ ' ( z )Figure . H.6shows how it cou to adapt the cascade of A ( z ) and B ( z ) .
A CASCADE OF NONLINEAR ADAPTIVE FILTERS
We next explore the question of how to train a cascade of adaptive nonlinear filte neural networks. The training method will be a form of backpropagation which similar role to that played by filtered-r LMS for the linear case. Consider thc cascade of nonlinear filters shown in Fig. H.7.In training th the objective is to adapt all the weights to minimize the mean square of the error. this becomes clear when the diagram of Fig. H.7 is redrawn as in Fig. H.8. The tw appear to be different but they function identically. The system of Fig. H.7is a cascade of two nonlinear filters, each a tapped connected to a two-layer neural network. The system of Fig. H.8, its equivalent nonlinear filter consisting of a tapped delay line connected to a four-layer neura The delays are z-' in the flow paths which bring signals to the inputs of the neura
d Desired response
Figure H.6
Training a cascade of A(z) and B(z) where B(z) is nonrninirnurn-pha
Since there are no delays within the network itself, the four layers can be adap backpropagation algorithm. Standard backpropagation will not be the correct algorithm, however. the system of Fig. H.8 to be equivalent to that of Fig. H.7, it is necessary tha of the weights of the first filter of Fig. H.7, which are labeled a , b, . . . ,i , be tra the weights of the first two layers of the network of Fig. H.8.This must be d as shown. Accordingly, all the weights of Fig. H.8 labeled a must have iden All the b weights must have identical values, and so forth. When adapting the the initial conditions must satisfy this restriction, and this restriction must be sa completion of each network adaptation cycle. A change in the a weights shou puted as the average charge that would be computed by conventional backpr there were no restrictions in the weights. A change in the b weights would be c like manner, and so forth. To adapt the cascade of Fig. H.7, one would compute weight changes w gram of Fig. H.8 and copy the corresponding weight values into the system of F represent symbolically the adaptation of a cascade of nonlinear filters like that by proper application of error backpropagation, the diagram of Fig. H.9 is used diagram could represent both SISO and MIMO systems.
H.5
NONLINEAR INVERSE CONTROL SYSTEMS BASED O NEURAL NETWORKS
Adaptation of cascaded nonlinear filters can be used in nonlinear inverse contro tive model-reference control system based on these principles is shown in Fig principle of operation is related to that of adaptive inverse control with the filt rithm. It is useful to compare the nonlinear system of Fig. H.10 with the linea Fig. 7.4. In Fig. 7.4, the overall system error is filtered through the plant inver
Figure H.7
I 1
z-'
A cascade of two nonlinear filters, each a tapped delay line connected to a two-layer neural network.
igure H.8
I
I
.m output
A system that is equivalent to that of Fig. H.7. It is a tapped delay line connected to a four-layer neural network.
Desired respo d
Figure H.9
A trainable cascade of nonlinear adaptive filters.
used to adapt the controller. Minimizing the mean square of this error optimiz troller. Having a precise plant inverse is not critical for the minimization of this
Command input
*
COPY controller
/
-
-
4 Plant emulator
Figure H.10 A model-reference adaptive inverse control system for a nonlinear plant. The plant the controller are tapped delay lines with neural networks connected to the taps.
In Fig. H. 10,the overall system error is backpropagatedthrough a direct m plant and then used to adapt the controller. The plant model or “emulator” adap backpropagation algorithm, is illustrated in Fig. H.10. The controller weights a train the cascade of Fig. H.9. This process is not used to traiii the “emulator co ever. Its weights are simply copied from those of the emulator itself. Having a pr emulator is not critical for the minimization of the overall system error.
model helicopter [ 11. In a much larger project, a full-sized self-driving van name (Autonomous Land Vehicle In a Neural Network) complete with video camera an onboard “brain” made from four workstations, has been developed and built a Mellon University [2]. ALVINN learned to drive by watching humans drive, an long distances at normal highway speeds, negotiating through traffic without h vention. The system is not yet perfect, of course, so when ALVINN drives, a h ways present to take over the controls if something goes wrong. In this section, we consider a system less complicated and more easily des ALVINN -that of a neural network which has learned to steer a computer-sim and trailer while backing to a loading platform. A solution to this highly nonlin problem was obtained by self-learning. The inputs to the two-layer network are “ ables: the angle and position of the rear of the trailer and the angle of the cab (see The output of the neural network is the angle of the steering wheel. The work w Nguyen and Widrow [3,4,5]. The learning algorithm they used, which is based o propagation algorithm, is called “backpropagation-through-time.” (Xcab, Ycab)
i
Figure H.ll
A
Truck,trailer, and loading dock.
The truck was only allowed to back up. Backing was done as a sequen steps. On the scale of a real 18-wheeler, each step would be a distance of appro meter. The truck backs from its initial position until it hits something and stops. T final state of the system involves having the rear of the trailer parallel to the loadi and positioned at its center. The actual final state is compared with the desired and the difference is a state error vector. After each backing-up sequence is com final error vector is used to modify the controller weights so that if the truck we
I
I
I
Figure H.12
Plant and controller.
-I I
I Zk+ I
I I
Neural-net controller I
I
Error vector
Figure H.13
6
Training the neural-net plant emulator.
The controller is also a two-layer neural network, trained as shown in Fig. initial position or state of the truck, 20 is applied to the controller which generat output, the steering wheel angle. Using this initial steering signal, U O , the truck step. The truck has now gone from the initial state 20 to the next state 21. The using the controller to set the steering angle and then backing a step is repeated the tryck hits something or the number of time steps exceeds a predetermined con final state of the run is Z K . The controller and emulator are each composed of two layers of adaptiv Every backing step is analogous to signals going through four layers. Backing fr
king
copy
p>1) uo
C = Controller
Figure H.14
T = Actual truck
T
...
E = Emulator of truck
Zk-I
Training the controller with backpropagation.
Final error for run
+
Zd
Desired final state
Adaptation of the emulator weights is done in accord with the scheme shown in After each backup sequence, the backpropagation-through-time algorithm dient of the squared positional error of the truck’s final state with respect to the the controller. This gradient is used to update the controller’s weights by stochas descent. Once learning is complete, the truck is able to back up satisfactorily from initial position, even “jackknifed,” and even from initial positions that were not encountered during training. The controller’s ability to react and respond reason positions is an example of generalization. An illustration of the functioning of trained system is shown in Fig. H.15. This is a laboratory exercise that could ture, have implications for vehicle control. More importantly, however, it serves demonstration of the capabilities of nonlinear networks. This demonstration he opment of the Intelligent Arc FurnaceTMcontroller described next.
APPLICATIONS TO STEEL M A K I N G
An electric arc furnace is used to melt and process scrap steel. The heat energy c a three-phase power line of rather massive capacity (often 30 megawatts or more electrical power for a city of 30,000people). The three-phase line connects to a b down transformers to supply current for three electrodes that stick down into t The electrodes made of graphite, are about 1 foot in diameter, and are about 2 Three independent servos control the depth of the electrodes into the furnace. When starting a new “heat,” scrap steel is loaded into the furnace and the activated to drive the electrodes down toward the scrap pile. When an arc is sparks fly and the noise is deafening. One’s first impression is that it’s like Dant Because the cost of installing and operating a large arc furnace is so great, changes in efficiency have a tremendous impact on economics. The motivation velopment of “intelligent control” is clear. In this section we describe the Inte FurnuceTMcontroller, invented by a recent Stanford graduate, Bill Staib of Neur tions Corporation [6]. The figures in this section were supplied by the inventor Figure H. 16 shows an arc furnace, its three-phase power system, and instr that provides signals useful for the control of the electrode servos. Currents an in the system are sensed, digitized, and fed to a 486 PC that implements the neu system. Numerical processing is performed by an 80 megaflop Intel i860 micr A microphone placed near the furnace provides the computer with the sounds Inferno. From all the sensed variables, a state vector is obtained.
I
End
Figure H.15
Example of a truck backup sequence.
Figure H. 17 shows the training of a neural network emulator of the furna ulator is used in the training of the controller or regulator, another neural net ure H.18 shows the training of the regulator. The learning algorithm is a variant propagation algorithm. It works in a similiu way to the training process for a of Fig. H.14 of the truck-backer. The results with neural control thus far have been excellent compared with systems that commonly exist for arc furnaces. Consumption of electric power is 5-8 percent, wear and tear on the furnace and the electrodes is reduced by about the power factor on the input power lines is brought closer to 1, and the daily th steel is increased by 10 percent. The neural controllers are being installed by plications Corporation just as quickly as they can be produced. These improv reportedly worth millions of dollars per year per furnace.
Figure H.16
Electric arc furnace and data acquisition system.
REG N+1
+
.
+ Neural-net
Figure H.17
Arc furnace emulator training.
Neural Network regulator
REG and state valuesfor 4 time N , N - 1
f
\*
-
REG(N + 1)
Neural-net furnace
-
D
Computer Technology Corporation), Austin, Texas, where the original work w 1989-1990 by John Havener of Texas Eastman and Jim Keeler of MCCiPavilion gies. In the original application conducted at the Texas Eastman Facility, Longv the neural network produced setpoint changes that reduced by one-third the requ an expensive chemical additive needed to remove byproduct impurities during p The facility produces plastics and chemical intermediates such as aldehydes a Since that work was complete, the technology and Pavilion’s Process Insights p been used in nearly 200 real-world applications, including modeling and optim distillation columns, modeling and control of plastics production, modeling and impurity levels in boilers, and so forth. These applicationshave generated treme backs, with savings of some applications totaling millions of dollars per year in a production facility. Texas Eastman, a division of Eastman Kodak, has been so sa the results that they are currently encouraging the use of neural networks throu Longview plant [8]. In making these applications, the first step is plant modeling or plant emul ically, the plant has many inputs (such as pressures, temperatures, flow rates, characteristics.) and one or more output parameters (such as yield, impurity le ance). The figures in this section were supplied by Pavilion. In Fig. H.19, an ada ral network is used to model an unknown plant, that is, to learn the plant’s dyna historical data. Once the plant emulator converges, it can be used to train the neural-net Figure H.20 shows how this is done. The error vector is the difference betwee output vector and the desired state vector. This error is backpropagated throug ral plant model to provide error signals for the adaptation of the weights of the The controller weights are adapted by the backpropagation algorithm to minimi of squares of the components of the error vector. Pavilion uses fuzzy logic in Insights package to establish constraints on some of the controlled variables. It is interesting to compare Figs. H. 14, H. 17, and H.20. Very similar thing on in the vehicle control system (the truck-backer),in the arc furnace control sys the chemical process control system. An emulator is made of the process to be and the controller is adapted by backpropagatingthe system error through the emu is a very powerful idea and it leads to useful applications. The reader should be a ever that this is not the only means of neural control. For instance, one method u Lord Corporation in the process of developing a new adhesive product [9] involv a neural network to model the product’s adhesive properties as a function of the proportions. After the network was trained, a search was performed over the n work’s input space to find the formulation believed by the network to be optim formulation was then fabricated and its adhesive properties were tested. This cre
F i g u r e H.19
Plant
Adaptive plant emulation.
\
\ \
,
Figure H.20
I
I
II
Using the plant model for training the controller by error backpropagatio
data point and the network was retrained to create a more accurate model. The input space was again searched, and this process was repeated until adequate re obtained. This section has described only a small fraction of the commercial and ind plications of neural networks that exist today. The list is long and impressive an rapidly. Before the turn of the century, we can reasonably expect to see neura become a household word and a part of everyday life. In Japan, fuzzy logic h achieved this status.
Bibliography for Appendix H
[ l ] T.J. PALLETT, and S. Ahmad, “Real-time neural network control of a min
copter in vertical flight,” in Proceedings of the 17th International Conferen plications of Art$cial Intelligence in Engineering - AIENGB2, Waterlo Canada, 1992, pp. 143-160.
[2] 0. PORT,“Sure, it can drive, but how is it at changing tires?” Business Week 1992, pp. 98-99.
[3] D.H. NGUYEN, and B. Widrow, “Neural networks for self-learning control IEEE Control Systems Magazine, Vol. 10, No. 3 (April 1990), pp. 18-23.
and B. WIDROW, “Neural networks for self-learningcontro [4] D.H. NGUYEN, Znt’l. J. Corrtrol, Vol. 54, No. 6 (1991), pp. 1439-1451.
Text Conventions
1. Boldfaced uppercase symbols denote column vectors or matrices. 2. The estimate of a scalar, vector, or matrix is designated by the use
placed over the pertinent symbol. I denotes the magnitude or absolute value of the scal within. The symbol 1 I 11 stands for the norm of the vector enclosed within. The inverse of a nonsingular matrix or the inverse of the transfer denoted by the superscript Transposition of a vector or a matrix is denoted by superscript T. Superscript plus (+) on a rational function of z denotes creation of a tion, which has all the poles and zeros of the original function which inside the unit circle in the z-plane. Superscript minus (-) on a rational function of z denotes creation of a tion, which has all the poles and zeroes of the original function which outside the unit circle on the z-plane. Superscript asterisk (*) denotes the optimal (or Wiener) solution. The asterisk (*) between two sequences indicates performing convo Transformation of a weight vector to the coordinates defined by th axes of the quadratic surface is denoted by superscriptprime ('). Subscript denotes that given impulse response is a causal on Subscript (a) denotes the delayed version (e.g., delayed inverse). Subscript c o p y indicates that the weights of the given filter are co the filter computed by an adaptive algorithm at another part of the s Subscript OPT means the optimal value of the parameter. Time reversal may be denoted by putting (-) over the pertinent sym Symbols truce ( ) and tr stand for taking trace of the matrix within thesis or immediately to the right. Taking gradient is denoted by (V). Summation operator is denoted by (C).
3. The symbol 1 4. 5. 6. 7.
8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
-'.
then the numerator of the expression is taken. 25. When an expression or a symbol of an expression is used as a sub the denominator of the expression is taken. 26. The symbol diag [A,,. . .,A,,, . . . ,A,,] denotes a diagonal matrix who on the main diagonal are A!,. . , ,A,,, ... ,An. 27. The symbol << denotes that the expression on the left of the sym smaller than the one on the right. 28. Signal vectors and transfer function matrices of MIMO systems are I. by 29. Symbol denotes definition. 30. Symbol =Z denotes approximate equality. 31. In constructing block diagrams, the following symbols are used. Th
A
++
denotes an adder with c = Q lows
b
+ b. The same symbol with algebraic s
2
denotes a subtractor with c = ab. The symbol
stands for multiplicative weight. Number inside the circle (if presen the value of the weight. Arrow through the weight denotes that we with a variable weight. The symbol
-0-
denotes an analog filter or a digital filter. An arrow through such a b that the impulse response is being adapted. The arrow can be conn signal or not connected at all. In the first case, the signal is used input used to adapt the filter. In the second case, actual adaptation is in another part of the system. Then the impulse response is copied in under consideration. 32. Operator z denotes z-transform.
DCT DFT DSD EPS FIR GEN Hg IIR IV LMS LRS MAX MIMO MIN MSE RLS SAN SISO SLAC TLU
Digital Cosine Transform Digital Fourier Transform Differential Steepest Descent Earphone Signal Finite Impulse Response (Noise) Generator Mercury Infinite Impulse Response Intravenous Least Mean Square Linear Random Search Maximum Multiple-Input Multiple-Output Minimum Mean Square Error Recursive Least Squares Algorithm Strong Ambient Noise Single-Input Single-Output Stanford Linear Accelerator Center Threshold Logic Unit
transfer function of a controller controller weight vector desired response at instant k digital signal at instant k transfer function of fk feedback controllers output of a digital filter at time k z-transform of a sequence gk impulses of impulse responses transfer function of a plant or a digital filter transfer function from point A to point B transfer function of a MIMO system from input i to output j command input at time instant k identity matrix z-transform of command input input controller square root of - 1 index time index number of inputs and outputs of a MIMO sys index “source” noise at time instant k z-transform of noise l k number of weights in the controller index misadjustment model reference microphones number of weights in the adaptive plant mode noise at instant k equivalent plant noise at sample k data length noise vector the order of kth impulse of plant impulse response plant weight vector crosscorrelation vector
autocorrelation matrix digital signal at sample k squaring device plant input at time k controller output at time k transfer function of U k and u; vectors of U k and u; variance of weight noise vector vector of weight errors initial weight error vector weight number I weight number I at time instant k weight vector input signal at time instant k input number 1 at time instant k input vector transform of signal Xk matrix of vectors x k plant dynamic response at sample k z-transform of signal Y k , digital filter transfer function plant output at time instant k location of a pole of a rational function individual weight noise variance location of a pole of a rational function coefficients of a polynomial. dither signal at sample k Kronecher delta delay vector of dither inputs hk errors in plant model and controller error dynamic system error ideal error noise transform eigenvalues diagonal matrix of eigenvalues
z-transforms of autocorrelation functions z-transforms of crosscorrelation functions time constants time constant of mean square error infinity
A Adaline, 61, 304,410,415 Adaptation: fast, 59 slow, 59 Adaptive: algorithms: efficiency,74-77 controls, 1 echo canceler, 4 10 equalization, 410 filter, 339 causal, 60 FIR, 60 nonlinear, 303, 305, 306, 475, 477, 479-484 filtering, I , 59 inverse control, see Inverse control linear bypass, 304 linear combiner, 61, 339,414 noise canceling, 59.77-81 plant modeling, see Plant, identification predictor, 7 1-74 resonance theory (ART) 1-3.410 signal processing, 1 threshold element, see Adaline Aileron, 235 control signals, 235 Aircraft noise canceling, 296-301 Aircraft vibrational control, 27,234-235 Astrom and Wittenmark, 363-368 Audio input signal, 184 Autocorrelation,42,48, 350, 384, 39 1 Toeplitz, 391
B Backpropagation, 29, 33, 304, 328,409464, 475,493
through time, 457,484 Beam control, 32 Bias, 249 Bias weight, 416 Blood pressure control, 26, 154Boltzmann machine, 410 Bootstrapping, 410 Bowl-shaped, 62 Bunch trajectory, 399
C
Canard wings, 235 Characteristic polynomial, 224 Chemical process industry, 491-4 CMAC, 410 Command input, 2, 188,262 Control: effort, 147, 148 nonlinear: use of neural networks, 3 Controller, 2 4 , 138, 262, 485, 4 erroneous, 262 error in, 262 ideal, 319 output power, 283 weights: noise variance, I78 Convergence: conditions, 349 of mean, 66,344,353 of variance, 66,180,206,20 353 speed, 383 Convolution, 41 Covariance matrix, 342,344 Crosscorrelation, 43,62, 384
DFTILMS algorithm, see LMS, DFTILMS algorithm Differential steepest-descent (DSD), 139,31 I Digital filter: causal, 43 linear. 40,41 noncausal, 43 tapped delay line, see Digital filter, transversal transversal, 4 1, 89 weights, 40 Discrete Hartley transform, 386 Disturbance, 1, 2, 5 , 89, 188, 262, 31 1, 319, 320,369 canceler, 258 insensitivity, 254-255 optimality, 2 12-2 15 canceling, 13, 59, 209-257, 264, 328, 372-38 1 MIMO, 290-292 nonlinear, 303,319-321 nonlinear MIMO, 320-321 optimal, 321 stability, 223-226 cance!ing loop, 262 minimizing, 212 model, 212 power, 378 ramp, 239,246 reference, 2 11 RMS, 400 spectrum, 249 step, 239,246 uncanceled, 215, 228, 256, 258, 260, 261,265 white, 212 Dither, 29, 93-97, 307, 311, 313, 328, 349362 noise, 141, 143, 144, 163,258,260,265, 266,287 optimal, 159
scheme C, 30, 94, 108, 1 301,309,310,328,34 MIMO modeling with, scheme C N L309-311,31 , signal, 90 white, 223,227,229 Dynamic response distortion, 26 Dynamic system error, 128,287
E
Earphone nose cancelation, 236Eigenvalue, 182,206, 223,383 largest, 64 matrix, 63 spread, 194, 197,359,383 Eigenvector: matrix, 63 Einstein, 398 Electric arc furnace, 487,490 Electrocardiography, 79 adult and fetal, 59, 79 Envelope: exponential, 64 Equalization, see Inverse modeli Error correction rules, 428437 linear rules, 429430 nonlinear rules, 430-434 Error signal, 2, 276
F
Feedback, 4,27 1 beam steering, 400 disturbance canceling, 209, filter, 211, 264 stabilizer, 2.3 1, 369, 377-3 unity, 2 Feedfonvard: filtering, 21 1 network, 421-424 Filter: bandpass, 389
G
Gaussian elimination, 248 Geometric: ratios, 34 I solution, 64 Gradient, 65 estimated, 65 estimation, 83 instantaneous, 66, 304 measured, 65 noise, 65.67-71 covariance, 76 unbiased estimate, 66,67 vector, 62, 64
H Havener, John, 49 1 Hebbian learning, 410 Hyperellipsoid, 39 1 Hyperplane, 4 16
I Idealized modeling, 90-91 IEEE Transactions on Neural Networks, 304 IIR, 89 Impulse response, 41, 337, 338 noncausal, 42 optimal, 60 Input correlation matrix R,62 Input signal X, 62 Instability, see Plant, instability adaptive process, 201 Intelligent control systems, 33 Inverse, 272 control, 257 model-reference, 4, 26 nonlinear, 303-328, 480484 modeling, 2 nonlinear, 303, 313 plant model, 6
K
Keeler, Jim, 491
1
Law of large numbers, 354 Learning curve, 67 Linearly separable logic functio Linear accelerator, 397 Linear classifier, 419 capacity, 417-419 Linear quadratic Gaussian (LQ 400,404 Linear random search (LRS), 13 LMS: a-LMS algorithm, 4 2 9 4 3 algorithm, 25, 59, 65, 32 352,383,409,414 DCTILMS algorithm, 3 249,383-394 robustness, 393 DFTILMS algorithm, 3 1,3 robustness, 393 filtered+, 27, 28, 160, 16 180, 197-201, 285 296-301,312-315,3 filtered-X, 27, 160-165, 18 197,201,207 problems, 188-1 94 filters: adaptive, 59-84 p L M S algorithm, 4 4 W 3 spectrum analyzer, 393
M
Madaline, 33,409464 Markov process: first-order, 8, 14, 188, 392-3 Matrix inversion lemma, 384, 38 Matrix multiplication, 270 Mean square error, see MSE Memoryless, 323
279,285,301,342-345,349,351, 356-357 Mismatch, 89.91-97, 102 sources, 89-90 Model-reference control, see Inverse, control, model-reference Modeling signal, 141 Modem, 410 MSE, 62,283,346 average, 73 excess, 346,355,357 minimum, 57, 174.3 19,339,355 small-sample-size,84 surface, 4 3 8 4 , 4 4 7 , 4 6 1 4 6 2
N Neural Applications Corporation, 487 Neural control systems, 33 Neural network, 61, 311,328,409,477,480484,491493 training algorithms, 409 backpropagation, see Backpropagation backpropagation-through-time, see Backpropagation, through time LMS algorithm, see LMS,algorithm Madaline rule I, 409, 411, 421, 434436 Madaline rule 11,409, 41 1, 436437, 460
Madaline rule III,409,411,44547, 458461 Perceptron rule, 409.43 1 4 3 3 Neural Networks, 304 Noise: canceler, 183-1 86 multiple-reference, 79 in weight vector, 83 reference, 77,211 synthetic, 215, 321 white, 48
inverse modeling, 125, 14 Online: adaptation, 322,328 inverse modeling, 125,29 Orthogonalizing algorithms, 3 1 Orthonormal modal matrix, 63 Output, 89 distortion, 264 disturbance power, 256, 2 error power, 228,256,258 min, 228 noise power, 260, 268 signal distortion, 227 transform, 227 Overall system performance, 25
P
Panic button, 21 I , 258 Perceptron, 33, 409-464 Persistently exciting, 30, 90 Plant, 89 delayed inverse, 166 discretized equivalent, 7 discretized stabilized, 14 disturbance, see Disturban controlling, 149 dynamic response, 89 emulator, 483,485,490,4 equivalent, stable, 3 1, 111 identification,2,3,59,88,97 221,223,328,349,3 imperfect, 262 MIMO, 274285,301 nonlinear, 307,323 stability, 223-226 input signal: power, 256 instability, 1-3 inverse, 3 delayed, 115,290,2% nonlinear, 328
modeling, see Plant, identification nonlinear, 303, 319, 323 nonminimum-phase,2,4, 13, 202,266268 nonstationary, 3 stabilized, 370-372 unstable, 3 1, 369-382 Poles: outside unit circle of 2-plane, 4 right half of s-plane, 4 Polynomial discriminant function, 419421 Positive definite, 63 Positive semidefinite, 63 Power normalization, 388, 390, 391 Primary input, 77 Primed coordinates, 64 Principal axes, 64 Punishheward, 410
Q Quadratic function, 62, 304 Quasistatic stationarity,4
R Rate of adaptation, 64 Rate of convergence, 59 filtered-€ LMS, 170-175 Recursive least squares (RLS) algorithm, 31, 77, 194,383-394 Reference model, 4, 13, 117, 188, 262 Reflection coefficients, 384 Reinforcement learning, 410 Rosenblatt, 304
S Sampling rate, 264 Self-tuning regulator, 30, 363-368 Sensor noise, 5,89 Sequential regression (SER) algorithm, 405 Servo, 235
Stability, 2,64, 230, 256, 258, 2 285,339-342,383 criteria, 226,261 filtered-€LMS, 170-175 LMS filter, 29,59 of variance, 357 range of p, 182, 206, 25 341,344,350,360 Staib, Bill, 487 Stanford Linear Accelerator Cen 407 Steel making, 487488 Steepest descent: method, 62,64,65, 304 rules, 43745 1 linear, 4 3 U 3 nonlinear rules, 44345 Strong ambient noise, 184 System error, 201 dynamic, 143, 144 variance, 175 variance of, 179, 181, 1 288 excess, 143,262 overall, 141, 143, 159, 289 truncation, 143 System integration, 27, 257-269 MIMO. 292-296
T
Tapped delay line, 60,326, 339 Threshold logic functions, 416 Threshold logic unit, see Adaline Time constant, 65, 67, 108, 17 223, 228,255,258, 26 339-342,350,358,36 large, 261 Time variant, see Plant, nonstatio Tracking ability, 386 Trajectory control, 399
Unit circle, 256, 337, 338 Unit delay, see Delay, unit
V Variable weights, 60 Variance: of dynamic system error, 136,264 of output error, 264 Vector difference equation, 64 Vector error signal, 65 Volterra: filter, 304,307,326 model, 309 series, 304
W Walsh-Hadamard transform, 386 Weight-vector: controller, 138 noise, 67-69, 358, 359
Widrow-Hoff LMS algorithm, s rithm Widrow and Hoff, 304 Wiener: disturbance canceler, 214 filter, 24,40-57 causal, 53, 57, 372 noise filter, 47 noncausal, 40,45,57,9 two-sided, see Wiener causal filtering, 25 solution, 63.84, 113,339 theory, 25 Wiener-Hopf equation, 46,57,2 causal, 5 1 matrix form, 63
z z-transform, 41